Grid size selection for nonlinear least-squares optimisation in spectral estimation and array processing

Abstract

In many spectral estimation and array processing problems, the process of finding estimates of model parameters often involves the optimisation of a cost function containing multiple peaks and dips. Such non-convex problems are hard to solve using traditional optimisation algorithms developed for convex problems, and computationally intensive grid searches are therefore often used instead. In this paper, we establish an analytical connection between the grid size and the parametrisation of the cost function so that the grid size can be selected as coarsely as possible to lower the computation time. Additionally, we show via three common examples how the grid size depends on parameters such as the number of data points or the number of sensors in DOA estimation. We also demonstrate that the computation time can potentially be lowered by several orders of magnitude by combining a coarse grid search with a local refinement step.

Fast fundamental frequency estimation: Making a statistically efficient estimator computationally efficient

“…• From a computational perspective, it is much more efficient to evaluate the cost function on a coarse grid followed by a refinement step than to evaluate the cost function on only a fine grid [26,27]. The coarseness of the grid is controlled with the scalar g, and we will find a suitable value for it in the Sec 6.…”

Loudspeaker and Listening Position Estimation Using Smart Speakers

“…The described signal model and estimation procedure can be used for any array geometry. In the experiments, we have used a uniform circular array (UCA) since the DOA estimation performance is independent of the direction of the source [15,16] and fast estimation algorithms for it exist [17]. Moreover, a UCA is often used in smart speakers.…”

A Fast Algorithm for Maximum-Likelihood Estimation of Harmonic Chirp Parameters

“…We use the approach in [28] to identify an appropriate grid size and start with a second order Taylor approximation of a function f : R → R around a local maximum with no active constraintsx such that the derivative is f (x) = 0. The objective function f (x) has approximately decreased by a factor of g from the value f (x) when x is given by…”

“…For the harmonic model the selection F = 5N L was also discussed and justified in [28], [29]. In Sec.…”

“…In this section, we evaluate MATLAB implementations of the presented algorithms in terms of accuracy and computational savings by Monte Carlo simulations and timing benchmarks 3 . In particular, we will investigate the behavior of the NLS method and empirically identify a usable value for K, with fixed F = 5N L as suggested in [28], [29]. Notice that we are not proposing any new method but simply improving upon the computational complexity of the known NLS method so we also refer to [10], [11] for additional assessments.…”