Jesper Kjær Nielsen

Default Bayesian Estimation of the Fundamental Frequency

Joint fundamental frequency and model order estimation is an important problem in several applications. In this paper, a default estimation algorithm based on a minimum of prior information is presented. The algorithm is developed in a Bayesian framework, and it can be applied to both real- and complex-valued discrete-time signals which may have missing samples or may have been sampled at a non-uniform sampling frequency. The observation model and prior distributions corresponding to the prior information are derived in a consistent fashion using maximum entropy and invariance arguments. Moreover, several approximations of the posterior distributions on the fundamental frequency and the model order are derived, and one of the state-of-the-art joint fundamental frequency and model order estimators is demonstrated to be a special case of one of these approximations. The performance of the approximations are evaluated in a small-scale simulation study on both synthetic and real world signals. The simulations indicate that the proposed algorithm yields more accurate results than previous algorithms. The simulation code is available online.

On compressed sensing and the estimation of continuous parameters from noisy observations

Compressed sensing (CS) has in recent years become a very popular way of sampling sparse signals. This sparsity is measured with respect to some known dictionary consisting of a finite number of atoms. Most models for real world signals, however, are parametrised by continuous parameters corresponding to a dictionary with an infinite number of atoms. Examples of such parameters are the temporal and spatial frequency. In this paper, we analyse how CS affects the estimation performance of any unbiased estimator when we assume such infinite dictionaries. We base our analysis on the Cramer-Rao lower bound (CRLB) which is frequently used for benchmarking the estimation accuracy of unbiased estimators. For the popular sensing matrices such as the Gaussian sensing matrix, our analysis shows that compressed sensing on average degrades the estimation accuracy by at least the down-sample factor.

The single- and multichannel audio recordings database (SMARD)

A new single- and multichannel audio recordings database (SMARD) is presented in this paper. The database contains recordings from a box-shaped listening room for various loudspeaker and array types. The recordings were made for 48 different configurations of three different loudspeakers and four different microphone arrays. In each configuration, 20 different audio segments were played and recorded ranging from simple artificial sounds to polyphonic music. SMARD can be used for testing algorithms developed for numerous application, and we give examples of source localisation results.

Bayesian Model Comparison With the g-Prior

Model comparison and selection is an important problem in many model-based signal processing applications. Often, very simple information criteria such as the Akaike information criterion or the Bayesian information criterion are used despite their shortcomings. Compared to these methods, Djurics asymptotic MAP rule was an improvement, and in this paper, we extend the work by Djuric in several ways. Specifically, we consider the elicitation of proper prior distributions, treat the case of real- and complex-valued data simultaneously in a Bayesian framework similar to that considered by Djuric, and develop new model selection rules for a regression model containing both linear and non-linear parameters. Moreover, we use this framework to give a new interpretation of the popular information criteria and relate their performance to the signal-to-noise ratio of the data. By use of simulations, we also demonstrate that our proposed model comparison and selection rules outperform the traditional information criteria both in terms of detecting the true model and in terms of predicting unobserved data. The simulation code is available online.

Bayesian Interpolation and Parameter Estimation in a Dynamic Sinusoidal Model

In this paper, we propose a method for restoring the missing or corrupted observations of nonstationary sinusoidal signals which are often encountered in music and speech applications. To model nonstationary signals, we use a time-varying sinusoidal model which is obtained by extending the static sinusoidal model into a dynamic sinusoidal model. In this model, the in-phase and quadrature components of the sinusoids are modeled as first-order Gauss-Markov processes. The inference scheme for the model parameters and missing observations is formulated in a Bayesian framework and is based on a Markov chain Monte Carlo method known as Gibbs sampler. We focus on the parameter estimation in the dynamic sinusoidal model since this constitutes the core of model-based interpolation. In the simulations, we first investigate the applicability of the model and then demonstrate the inference scheme by applying it to the restoration of lost audio packets on a packet-based network. The results show that the proposed method is a reasonable inference scheme for estimating unknown signal parameters and interpolating gaps consisting of missing/corrupted signal segments.

Bayesian model comparison and the BIC for regression models

In the signal processing literature, many methods have been proposed for solving the important model comparison and selection problem. However, most of these methods only find the most likely model or only work well under particular circumstances such as a large number of data points or a high signal-to-noise ratio (SNR). One of the most successful classes of methods is the Bayesian information criteria (BIC) and in this paper, we extend some of the recent work on the BIC. In particular, we develop methods in a full Bayesian framework which work well across a large/small number of data points and high/low SNR for either real- or complex-valued data originating from a regression model. Aside from selecting the most probable model, these rules can also be used for model averaging as they assign a probability to each candidate model. Through simulations on a polynomial trend model, we demonstrate that the proposed rules outperform other rules in terms of detecting the true model order, de-noising the noisy signal, and making predictions of unobserved data points. The simulation code is available online.

On frequency domain models for TDOA estimation

Time-difference-of-arrival (TDOA) estimation is an important problem in many microphone signal processing applications. Traditionally, this problem is solved by using a cross-correlation method, but in this paper we show that the cross-correlation method is actually a restricted special case of a much more general method. In this connection, we establish the conditions under which the crosscorrelation method is a statistically efficient estimator. One of the conditions is that the source signal is periodic with a known fundamental frequency of 2π/N radians per sample, where N is the number of data points, and a known number of harmonics. The more general method only relies on that the source signal is periodic and is, therefore, able to outperform the cross-correlation method in terms of estimation accuracy on both synthetic data and artificially delayed speech data. The simulation code is available online.

An approximate Bayesian fundamental frequency estimator

Joint fundamental frequency and model order estimation is an important problem in several applications such as speech and music processing. In this paper, we develop an approximate estimation algorithm of these quantities using Bayesian inference. The inference about the fundamental frequency and the model order is based on a probability model which corresponds to a minimum of prior information. From this probability model, we give the exact posterior distributions on the fundamental frequency and the model order, and we also present analytical approximations of these distributions which lower the computational load of the algorithm. By use of simulations on both a synthetic signal and a speech signal, the algorithm is demonstrated to be more accurate than a state-of-the-art maximum likelihood-based method.

Joint direction-of-arrival and order estimation in compressed sensing using angles between subspaces

In this paper, we consider the problem of joint direction-of-arrival and order estimation in array processing with compressed sensing. In particular, we show how to solve these problems jointly using a subspace approach based on the notion of angles between subspaces. In the process, we also discuss the conditions on the measurement matrix and demonstrate how to implement the estimator algorithm efficiently when using compressed sensing. Our simulation results show that it is indeed possible to solve these problems and that good performance can be obtained, although the use of compressed sensing does have an impact on the performance of the estimator.

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

Modelling signals as being periodic is common in many applications. Such periodic signals can be represented by a weighted sum of sinusoids with frequencies being an integer multiple of the fundamental frequency. Due to its widespread use, numerous methods have been proposed to estimate the fundamental frequency, and the maximum likelihood (ML) estimator is the most accurate estimator in statistical terms. When the noise is assumed to be white and Gaussian, the ML estimator is identical to the non-linear least squares (NLS) estimator. Despite being optimal in a statistical sense, the NLS estimator has a high computational complexity. In this paper, we propose an algorithm for lowering this complexity significantly by showing that the NLS estimator can be computed efficiently by solving two Toeplitz-plus-Hankel systems of equations and by exploiting the recursive-in-order matrix structures of these systems. Specifically, the proposed algorithm reduces the time complexity to the same order as that of the popular harmonic summation method which is an approximate NLS estimator. The performance of the proposed algorithm is assessed via Monte Carlo and timing studies. These show that the proposed algorithm speeds up the evaluation of the NLS estimator by a factor of 50–100 for typical scenarios.