FM Parameter Estimation with Low-Order Rational Constraints on Wasserstein Loss Landscape
Frequency modulation (FM) synthesis has been widely used in music production and sound design due to its ability to generate rich timbres with few control parameters. However, estimating the frequency parameters from a target sound remains challenging because different parameter configurations can yield similar spectra, creating numerous local minima in the loss landscape. In this paper, we analyze the Wasserstein distance loss landscape for two-operator FM synthesis under practical FFT-based spectral representations and show that it exhibits non-differentiable ridges at rational frequency ratios, arising from negative-frequency folding and spectral ordering transitions. Exploiting this structure, we propose a constrained gradient-based optimization strategy that constrains the frequency ratio in each optimization run to an interval bounded by consecutive low-order rational ratios and retains the lowest-loss candidate across intervals. Experimental results from controlled ablations show that maintaining the constraint throughout optimization improves reliability over random initialization and initialization-only constraints, particularly for more complex spectra at higher modulation indices.