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Comment by Lichtso | Hacker News Reader

Also try using the Wavelet-Transform instead of short time FFT (with overlapping windows).

It is easier to configure (less parameters, there is no need for a window function), offers more flexibility (exponential frequency band; e.g. for music scales) and can reach the Gabor-Heisenberg uncertainty limit without artifacts.

The only downside is that you need to know the entire signal in advance, so it can only be used for recordings.

Shameless self-promo of my implementation: https://github.com/Lichtso/CCWT

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Shameless cross-promo; ported your implementation to JS https://github.com/grz0zrg/ccwt.js

Your repo and tutorial are really cool! But do you have any sort of interactive "out-of-the-box" demo that doesn't require me to write code to call the library (online or downloadable)?

I made a WASM + WebGL port some time ago, similar to this one here. I can polish and upload it to GitHub in the next days.

You don't need to know the entire signal in advance - you can just window the wavelets at some reasonable size.

Yes, but then you are back at time-windows, a window-function, overlap and artifacts, which is defeating the purpose.

I'm more familiar with Fourier transforms and have limited experience with wavelets. But if each wavelet intrinsically falls off at a Gaussian curve, cutting it off (possibly with a window) at 3-4 sigmas won't change the wavelet substantially. Maybe for some use cases, the wavelet will be narrower at high frequencies (short delay), and wider at low frequencies (high delay). I don't know how you'd perform incremental updates of a plot drawn with non-uniform delays though...

The signal will continue over the seam between two windows, meaning you will cut the wave in the signal "in half". Mathematically, waves are always infinite and to cut them you would actually introduce overtones (higher frequencies) to model the sharp end / start of the base wave. These then result in artifacts regardless of what method is used for the transformation (Fourier or Wavelet).

> The signal will continue over the seam between two windows, meaning you will cut the wave in the signal "in half".

You can window the wavelet, then slide the finite-duration wavelet by a few samples at a time, even if the wavelet is hundreds to thousands of samples long. This is possible in STFT as well (each part of the original signal shows up in many separate FFTs).

Again, I don't know the implementation details of wavelet transforms. Maybe I'll look into your repo when I have time. What's your asymptotic and practical runtime?

O(n log n) for the x axis (time samples) and O(n) for the y axis (frequencies).

But you can downsample the signal in frequency domain, meaning you will pay mostly for the output resolution.

You often want a wavelet with compact support anyway so you don't even need to choose the window size: it's built-in.

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