  Machine learning approaches to modelling analog audio effects have seen
intensive investigation in recent years, particularly in the context of
non-linear time-invariant effects such as guitar amplifiers. For modulation
effects such as phasers, however, new challenges emerge due to the presence of
the low-frequency oscillator which controls the slowly time-varying nature of
the effect. Existing approaches have either required foreknowledge of this
control signal, or have been non-causal in implementation. This work presents a
differentiable digital signal processing approach to modelling phaser effects
in which the underlying control signal and time-varying spectral response of
the effect are jointly learned. The proposed model processes audio in short
frames to implement a time-varying filter in the frequency domain, with a
transfer function based on typical analog phaser circuit topology. We show that
the model can be trained to emulate an analog reference device, while retaining
interpretable and adjustable parameters. The frame duration is an important
hyper-parameter of the proposed model, so an investigation was carried out into
its effect on model accuracy. The optimal frame length depends on both the rate
and transient decay-time of the target effect, but the frame length can be
altered at inference time without a significant change in accuracy.
