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Title: DDSP: Differentiable Digital Signal Processing

Paper content:
Most generative models of audio directly generate samples in one of two domains: time or frequency.
While sufficient to express any signal, these representations are inefficient, as they do not utilize existing knowledge of how sound is generated and perceived.
A third approach (vocoders/synthesizers) successfully incorporates strong domain knowledge of signal processing and perception, but has been less actively researched due to limited expressivity and difficulty integrating with modern auto-differentiation-based machine learning methods.
In this paper, we introduce the Differentiable Digital Signal Processing (DDSP) library, which enables direct integration of classic signal processing elements with deep learning methods.
Focusing on audio synthesis, we achieve high-fidelity generation without the need for large autoregressive models or adversarial losses, demonstrating that DDSP enables utilizing strong inductive biases without losing the expressive power of neural networks.
Further, we show that combining interpretable modules permits manipulation of each separate model component, with applications such as independent control of pitch and loudness, realistic extrapolation to pitches not seen during training, blind dereverberation of room acoustics, transfer of extracted room acoustics to new environments, and transformation of timbre between disparate sources.
In short, DDSP enables an interpretable and modular approach to generative modeling, without sacrificing the benefits of deep learning.
The library will is available at https://github.com/magenta/ddsp and we encourage further contributions from the community and domain experts.
Neural networks are universal function approximators in the asymptotic limit (Hornik et al., 1989) , but their practical success is largely due to the use of strong structural priors such as convolution (LeCun et al., 1989) , recurrence (Sutskever et al., 2014; Williams & Zipser, 1990; Werbos, 1990) , and self-attention (Vaswani et al., 2017) .
These architectural constraints promote generalization and data efficiency to the extent that they align with the data domain.
From this perspective, end-to-end learning relies on structural priors to scale, but the practitioner's toolbox is limited to functions that can be expressed differentiably.
Here, we increase the size of that toolbox by introducing the Differentiable Digital Signal Processing (DDSP) library, which integrates interpretable signal processing elements into modern automatic differentiation software (TensorFlow) .
While this approach has broad applicability, we highlight its potential in this paper through exploring the example of audio synthesis.
Objects have a natural tendency to periodically vibrate.
Small shape displacements are usually restored with elastic forces that conserve energy (similar to a canonical mass on a spring), leading to harmonic oscillation between kinetic and potential energy (Smith, 2010) .
Accordingly, human hearing has evolved to be highly sensitive to phase-coherent oscillation, decomposing audio into spectrotemporal responses through the resonant properties of the basilar membrane and tonotopic mappings into the auditory cortex (Moerel et al., 2012; Chi et al., 2005; Theunissen & Elie, 2014) .
However, neural synthesis models often do not exploit this periodic structure for generation and perception.
The DDSP library fuses classical DSP with deep learning, providing the ability to take advantage of strong inductive biases without losing the expressive power of neural networks and end-to-end learning.
We encourage contributions from domain experts and look forward to expanding the scope of the DDSP library to a wide range of future applications.
A APPENDIX Figure 5 : Decomposition of a clip of solo violin.
Audio is visualized with log magnitude spectrograms.
Loudness and fundamental frequency signals are extracted from the original audio.
The loudness curve does not exhibit clear note segmentations because of the effects of the room acoustics.
The DDSP autoencoder takes those conditioning signals and predicts amplitudes, harmonic distributions, and noise magnitudes.
Note that the amplitudes are clearly segmented along note boundaries without supervision and that the harmonic and noise distributions are complex and dynamic despite the simple conditioning signals.
Finally, the extracted impulse response is applied to the combined audio from the synthesizers to give the full resynthesis audio.
Figure 6 : Diagram of the Additive Synthesizer component.
The synthesizer generates audio as a sum of sinusoids at harmonic (integer) multiples of the fundamental frequency.
The neural network is then tasked with emitting time-varying synthesizer parameters (fundamental frequency, amplitude, harmonic distribution).
In this example linear-frequency log-magnitude spectrograms show how the harmonics initially follow the frequency contours of the fundamental.
We then factorize the harmonic amplitudes into an overall amplitude envelope that controls the loudness, and a normalized distribution among the different harmonics that determines spectral variations.

assistant: text: Better audio synthesis by combining interpretable DSP with end-to-end learning.