# Fun with Wavelet Analysis

Not all the new functions in 2020.1 are specific to spacecraft. We have also been hard at work adding new functionality to the core toolbox. Here, I’d like to give an example of one of our new functions for performing a Wavelet analysis.

But what is a Wavelet analysis? Well, you plot the Wavelet transform of a signal when you want to visualize how the frequency spectrum changes in time. The Wavelet transform is a lot like a Fourier transform that you perform at every possible starting point, with an appropriate window function multiplied in so that you’re only looking at a portion of the signal.

But there’s one added wrinkle, because the frequency spectrum at a specific frequency at a specific time doesn’t technically exist. It’s not technically possible to know what the component of a signal at 100 kHz is at 0.5 seconds in, because the frequency spectrum depends on the entire signal. There has to be some trade-off between time resolution and signal resolution. If we look at a very long chunk of the signal, we can nail down its frequency components very well but we can’t see them change quickly. If we look at a very short chunk of the signal, we know precisely when the frequency changes but we can’t tell the difference between two similar frequencies. It’s a trade-off.

Now let’s get to the examples! The new function in 2020.1 is called WaveletMorlet because the specific window function we use is called the Morlet wavelet (A Wavelet transform using a Morlet wavelet is also called a Gabor transform). Here’s the signal that we’ll be analyzing:

We already know what we’re going to expect in this example. It looks like there’s a persistent, low-frequency component, then a higher-frequency component whose frequency goes up, peaks around 0.25 seconds, then goes down, and bottoms around 0.75 seconds. Here’s what the Wavelet transform looks like: