Nice! I've used a homegrown CQT-based visualizer for a while for audio analysis. It's far superior to the STFT-based view you get from e.g. Audacity, since it is multiresolution, which is a better match to how we actually experience sound. I have for a while wanted to switch my tool to a gammatone-filter-based method [1] but I didn't know how to make it efficient.

Actually I wonder if this technique can be adapted to use gammatone filters specifically, rather than simple bandpass filters.

[1] https://en.wikipedia.org/wiki/Gammatone_filter

I might be mistaking, but I don't see how this is novel. As far as I know, this has a proven DSP technique for ages, although it it usually only applied when a small amount of distinct frequencies need to be detected - for example DTMF.

When the number of frequencies/bins grows, it is computationally much cheaper to use the well known FFT algorithm instead, at the price of needing to handle input data by blocks instead of "streaming".

For some reason the value of Pi given in the C++ code is wrong!

It's given in the source as 3.14159274101257324219 when the right value to the same number of digits is 3.14159265358979323846. Very weird. I noticed when I went to look at the C++ to see how this algorithm was actually implemented.

https://github.com/alexandrefrancois/noFFT/blob/main/src/Res... line 31.

Just want to call out the resources listed at the bottom of the Resonate website:

- The Oscillators app demonstrates real-time linear, log and Mel scale spectrograms, as well as derived audio features such as chromagrams and MFCCs https://alexandrefrancois.org/Oscillators/

- The Resonate Youtube playlist features video captures of real-time demonstrations. https://www.youtube.com/playlist?list=PLVcB_ABiKC_cbemxXUUJX...

- The open source Oscillators Swift package contains reference implementations in Swift and C++.https://github.com/alexandrefrancois/Oscillators

- The open source python module noFFT provides python and C++ implementations of Resonate functions and Jupyter notebooks illustrating their use in offline settings. https://github.com/alexandrefrancois/noFFT

You can view this result as the convolution of the signal with an exponentially decaying sine and cosine.

That is, `y(t') = integral e^kt x(t' - t) dt`, with k complex and negative real part.

If you discretize that using simple integration and t' = i dt, t = j dt you get

    y_i = dt sum_j e^(k j dt) x_{i - j}
    y_{i+1} = dt sum_j e^(k j dt) x_{i+1 - j}
            = (dt e^(k dt) sum_j' e^(k j' dt) x_{i - j'}) + x_i 
            = dt e^(k dt) y_i + x_i

If we then scale this by some value, such that A y_i = z_i we can write this as

    z_{i+1} = dt e^(k dt) z_i + A x_i

Here the `dt e^(k dt)` plays a similar role to (1-alpha) and A is similar to P alpha - the difference being that P changes over time, while A is constant.

We can write `z_i = e^{w dt i} r_i` where w is the imaginary part of k

   e^{w dt (i+1)} r_{i+1} = dt e^(k dt) e^{w dt i} r_i + A x_i
             r_{i+1} = dt e^((k - w) dt) r_i + e^{-w dt (i+1) } A x_i
                     = (1-alpha) r_i + p_i x_i

Where p_i = e^{-w dt (i+1) } A = e^{-w dt ) p_{i-1} Which is exactly the result from the resonate web-page.

The neat thing about recognising this as a convolution integral, is that we can use shaping other than exponential decay - we can implement a box filter using only two states, or a triangular filter (this is a bit trickier and takes more states). While they're tricky to derive, they tend to run really quickly.