SINUSOIDAL FREQUENCY ESTIMATION BY GRADIENT DESCENT Ben Hayes Charalampos Saitis Gy orgy Fazekas Centre for Digital Music Queen Mary University of London

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SINUSOIDAL FREQUENCY ESTIMATION BY GRADIENT DESCENT

Ben Hayes, Charalampos Saitis, Gy¨

orgy Fazekas

Centre for Digital Music, Queen Mary University of London

ABSTRACT

Sinusoidal parameter estimation is a fundamental task in appli-

cations from spectral analysis to time-series forecasting. Esti-

mating the sinusoidal frequency parameter by gradient descent

is, however, often impossible as the error function is non-convex

and densely populated with local minima. The growing family of

differentiable signal processing methods has therefore been un-

able to tune the frequency of oscillatory components, preventing

their use in a broad range of applications. This work presents a

technique for joint sinusoidal frequency and amplitude estimation

using the Wirtinger derivatives of a complex exponential surro-

gate and any ﬁrst order gradient-based optimizer, enabling end-

to-end training of neural network controllers for unconstrained

sinusoidal models.

Index Terms—differentiable signal processing, machine

learning, sinusoidal parameter estimation

1. INTRODUCTION

Estimating sinusoidal parameters from a signal is a crucial step

in numerous signal processing algorithms, and a wealth of tech-

niques have been proposed in both the single and multiple sinu-

soid formulations. Most seek the maximum likelihood (ML) es-

timate of sinusoidal model parameters in the presence of white

Gaussian noise, the statistical properties of which are well estab-

lished [1].

Estimators of sinusoidal frequency must circumvent the non-

linearity of the model and non-convexity of the corresponding ob-

jective as a function of the frequency parameter. The most com-

mon approach is thus to apply a multi-stage algorithm in which

an initial frequency estimate is obtained through search heuris-

tics [2, 3, 4], spectral peak interpolation [5], discrete-time Fourier

transform (DTFT) decorrelation [6], or other procedures [7, 8, 9],

and then reﬁned using an optimization method. Alternate ap-

proaches include iteratively updating a model-based relaxation of

the problem [10, 8], linearizing the problem using delay operators

[11, 12], or deﬁning a surrogate model where an equivalence can

be drawn between solutions [13].

Such methods achieve accurate estimates but are unsuitable for

use in the context of end-to-end models ﬁt by gradient descent,

where integrating derivative-free operations or complex heuris-

tics is challenging and often unstable. In particular, the recent

Ben Hayes is supported by UK Research and Innovation [grant num-

ber EP/S022694/1]. This research utilised Queen Mary’s Apocrita HPC fa-

cility, supported by QMUL Research-IT. http://doi.org/10.5281/

zenodo.438045

proliferation of models applying differentiable digital signal pro-

cessing (DDSP) [14] – a family of techniques which allow neural

networks to directly control digital signal processors – highlights

the need for a method for sinusoidal frequency estimation by gra-

dient descent.

Applications of DDSP have included providing high level con-

trols for harmonic-plus-noise synthesizers [14], controlling digi-

tal synthesis methods with neural networks [15, 16], modelling

[17] and controlling [18] audio effects and direct ﬁlter design

[19]. Yet, despite success at these complex tasks, DDSP-based

models have so far been unable to predict sinusoidal frequency

parameters. Aspects of the problem have been acknowledged in

the literature. Turian & Henry [20] showed that frequency domain

distances lack a stable and informative frequency gradient, whilst

Engel et al. [21] used a parameter regression pretraining scheme

to circumvent issues with local minima when optimizing sinu-

soidal frequencies. Caspe et al. [16] similarly note that gradient

descent fails to tune the modulation frequencies of a differentiable

FM synthesizer due to ripple in the error function.

In this work, we propose a simple surrogate to the sinusoidal

oscillator with gradients that allow ﬁrst-order gradient based op-

timization. With this approach, we take a ﬁrst step towards end-

to-end learning of neural network controllers for a broader family

of differentiable audio synthesizers and signal processors.

2. SINUSOIDAL FREQUENCY ESTIMATION

We are concerned with modelling the class of discrete-time sig-

nals that can be expressed as:

xn=vn+X

k∈K

αkcos (ωkn+φk),(1)

where vn∼ N 0, σ2, and αk, ωk, φkare the amplitude, fre-

quency, and phase parameters, respectively, of unordered sinu-

soidal components with index set K⊆N. Following the stan-

dard ML derivations, ﬁnding estimates ˆαk,ˆωk,ˆ

φkis equivalent

to minimizing the mean squared error of the model. In many ap-

plications of machine learning to audio, we are concerned with

other formulations of the error. These can be accounted for by

expressing the likelihood in terms of other signal representations,

such as the discrete Fourier transform (DFT).

It is well established that when ωkand φkare known, this

problem is linear in αk[22] – a property which, for example, al-

lows DDSP models to directly predict harmonic amplitudes [14].

When φis unknown, an optimal estimate can be found by evaluat-

ing the DTFT at the known frequencies ωk. In the case where fre-

quency is unknown, however, the optimization problem is more

arXiv:2210.14476v2 [eess.SP] 18 Nov 2022

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SINUSOIDALFREQUENCYESTIMATIONBYGRADIENTDESCENTBenHayes,CharalamposSaitis,Gy¨orgyFazekasCentreforDigitalMusic,QueenMaryUniversityofLondonABSTRACTSinusoidalparameterestimationisafundamentaltaskinappli-cationsfromspectralanalysistotime-seriesforecasting.Esti-matingthesinusoidalfrequencyparameterbygradi...

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SINUSOIDAL FREQUENCY ESTIMATION BY GRADIENT DESCENT Ben Hayes Charalampos Saitis Gy orgy Fazekas Centre for Digital Music Queen Mary University of London

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