AM/FM Chirp Dataset Generation

We design a parametric sound synthesizer to generate a dataset of amplitude modulated chirp (AM/FM) signals, that is controlled by three parameters: carrier frequency (\(f_c\), in Hz), amplitude modulation frequency (\(f_m\), in Hz) and chirp rate (\(\gamma\), in octaves/second).

These are an idealization of the types of modulations found in musical instrument playing techniques.

We define a synthesizer \(\boldsymbol{g}\) of exponential ``chirps’’ with exactly three factors of variability: a carrier frequency \(f_c\), an amplitude modulation (AM) frequency \(f_m\), and a frequency modulation (FM) rate \(\gamma\).

Denoting by \(\boldsymbol{\theta}\) the triplet \((f_c, f_m, \gamma)\), we have for every \(\boldsymbol{\theta}\):

\[ \begin{equation} \boldsymbol{g_\theta}: t\longmapsto \boldsymbol{\phi}_{w}(\gamma t) \sin(2\pi f_\mathrm{m} t) \sin\left( \dfrac{2\pi f_{\mathrm{c}}}{\gamma \log 2} 2^{\gamma t} \right) \end{equation} \]

To ensure that we can visualize these 3 factors of acoustic variability on a 3D manifold embedding, we normalize the energy of the signals.

We restrict the bandwidth to exactly two octaves by varying the duration of a signal in inverse proportion to its chirp rate. This ensures that faster chirps do not cover a larger bandwidth over the same duration as a slow chirp.

We achieve this with \(\boldsymbol{\phi}_{w}\), a Gaussian window of characteristic width equal to \(w\).

The AM/FM signal \(\boldsymbol{g_\theta}\) has an instantaneous frequency of \(f_c 2^{\gamma t}\) and an essential duration of \(w/\gamma\). Thus, it covers a bandwidth \(w\), independently from \(\boldsymbol{\theta}\).

Importantly, in this task, there’s a benefit to the physical interpretability of scattering transforms.

We apply Eq. (\ref{eqn:synth}) for \(16\) values of \(f_c\), \(f_m\), and \(\gamma\), arranged in a geometric progression; hence yielding a dataset of \(16^3=4096\) audio signals in total.

We vary \(f_c\) between 512 Hz to 1024 Hz; \(f_m\), between and 4 Hz to 32 Hz; and \(\gamma\), between 0.5 and 4 octaves/second respectively.