- 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

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

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

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

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.

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".

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.

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 wrote this cross-disciplinary paper about resonance a few years ago. You may find it useful or at least interesting.

https://www.frontiersin.org/journals/neurorobotics/articles/...