Nonlinear Least Squares Methods for Joint DOA and Pitch Estimation

Abstract

In this paper, we consider the problem of joint direction-of-arrival (DOA) and fundamental frequency estimation. Joint estimation enables robust estimation of these parameters in multi-source scenarios where separate estimators may fail. First, we derive the exact and asymptotic Cramér-Rao bounds for the joint estimation problem. Then, we propose a nonlinear least squares (NLS) and an approximate NLS (aNLS) estimator for joint DOA and fundamental frequency estimation. The proposed estimators are maximum likelihood estimators when: 1) the noise is white Gaussian, 2) the environment is anechoic, and 3) the source of interest is in the far-field. Otherwise, the methods still approximately yield maximum likelihood estimates. Simulations on synthetic data show that the proposed methods have similar or better performance than state-of-the-art methods for DOA and fundamental frequency estimation. Moreover, simulations on real-life data indicate that the NLS and aNLS methods are applicable even when reverberation is present and the noise is not white Gaussian.

Joint Spatio-Temporal Filtering Methods for DOA and Fundamental Frequency Estimation

“…Moreover, a synthetic, multichannel periodic signal was used in every simulation with harmonics with unit amplitudes, and, in each simulation, the pitch and DOA were sampled from Hz Hz and , respectively, where denotes the continuous uniform distribution in the interval . The methods evaluated in these experiments are the optimal, white noise, and approximate filterbank ('bo', 'bw', and 'ba') and single filter ('so', 'sw', and 'sa') methods, the multichannel pitch estimator ('am') in [29], the steered response power method with phase transform ('sp') [30], and the exact and asymptotic nonlinear least squares (NLS) methods ('n' and 'an') in [10]. First, the performance of the proposed methods was evaluated for different 's in scenarios with 1) and an SNR of 30 dB, 2) and an SNR of 20 dB, and 3) and an SNR of 30 dB.…”

“…Another benefit is a potentially higher estimation accuracy. In [10], it was shown that the asymptotic Cramér-Rao bounds (CRBs) for the DOA and pitch are given by (13) (14) for the scenario described by (11) when is white noise and for , with PSNR denoting the pseudo signal-tonoise ratio. The PSNR is defined as (15) and is the variance of the noise.…”

“…This traditional and cascaded way of estimating the parameters will most likely increase the CRB of the parameter estimated in the second step of this procedure. The cause of this increase, is the linear transformation of the spatio-temporal data introduced by the signal extraction after estimation of the first parameter [10].…”

“…These have been found by first inserting the expressions in (21), (23), and (25) in (26), and then inserting (31), (33), and (34) in (35). We note that, except for a scaling factor of , the cost function for the approximative filterbank approach resembles that of the approximative NLS estimator in [10]. Furthermore, as mentioned in Section III-A, the above estimator can be used in an iterative algorithm to estimate the parameters of multiple sources [28].…”

“…It should also be noted that the DOA along with the pitch also are believed to be some of the governing factors that the human auditory system uses for separating sources. This line of reasoning has, quite recently, led to some joint DOA and fundamental frequency estimators, including maximum likelihood based [10], [11], subspace-based [12]- [14], correlation-based [15], [16], and filtering-based [17]- [19] methods. Notably, the problem of joint DOA and fundamental frequency estimation was formalized and thoroughly analyzed in [10], and a maximum likelihood estimator that achieves the highest possible accuracy (under certain conditions) was proposed.…”