Scott Rickard

Blind separation of speech mixtures via time-frequency masking

Binary time-frequency masks are powerful tools for the separation of sources from a single mixture. Perfect demixing via binary time-frequency masks is possible provided the time-frequency representations of the sources do not overlap: a condition we call W-disjoint orthogonality. We introduce here the concept of approximate W-disjoint orthogonality and present experimental results demonstrating the level of approximate W-disjoint orthogonality of speech in mixtures of various orders. The results demonstrate that there exist ideal binary time-frequency masks that can separate several speech signals from one mixture. While determining these masks blindly from just one mixture is an open problem, we show that we can approximate the ideal masks in the case where two anechoic mixtures are provided. Motivated by the maximum likelihood mixing parameter estimators, we define a power weighted two-dimensional (2-D) histogram constructed from the ratio of the time-frequency representations of the mixtures that is shown to have one peak for each source with peak location corresponding to the relative attenuation and delay mixing parameters. The histogram is used to create time-frequency masks that partition one of the mixtures into the original sources. Experimental results on speech mixtures verify the technique. Example demixing results can be found online at http://alum.mit.edu/www/rickard/bss.html.

Blind separation of disjoint orthogonal signals: demixing N sources from 2 mixtures

We present a novel method for blind separation of any number of sources using only two mixtures. The method applies when sources are (W-)disjoint orthogonal, that is, when the supports of the (windowed) Fourier transform of any two signals in the mixture are disjoint sets. We show that, for anechoic mixtures of attenuated and delayed sources, the method allows one to estimate the mixing parameters by clustering ratios of the time-frequency representations of the mixtures. The estimates of the mixing parameters are then used to partition the time-frequency representation of one mixture to recover the original sources. The technique is valid even in the case when the number of sources is larger than the number of mixtures. The general results are verified on both speech and wireless signals.

Comparing Measures of Sparsity

Sparsity of representations of signals has been shown to be a key concept of fundamental importance in fields such as blind source separation, compression, sampling and signal analysis. The aim of this paper is to compare several commonly-used sparsity measures based on intuitive attributes. Intuitively, a sparse representation is one in which a small number of coefficients contain a large proportion of the energy. In this paper, six properties are discussed: (Robin Hood, Scaling, Rising Tide, Cloning, Bill Gates, and Babies), each of which a sparsity measure should have. The main contributions of this paper are the proofs and the associated summary table which classify commonly-used sparsity measures based on whether or not they satisfy these six propositions. Only two of these measures satisfy all six: the pq-mean with p les 1, q > 1 and the Gini index.

On the approximate W-disjoint orthogonality of speech

It is possible to blindly separate an arbitrary number of sources given just two anechoic mixtures provided the time-frequency representations of the sources do not overlap, a condition which we call W-disjoint orthogonality. We define a power weighted two-dimensional histogram constructed from the ratio of the time-frequency representations of the mixtures which is shown to have one peak for each source with: peak location corresponding to the relative amplitude and delay mixing parameters. All of the time-frequency points which yield estimates in a given peak are exactly all the non-zero magnitude components of one of the sources. We introduce the concept of approximate W-disjoint orthogonality, present experimental results demonstrating the level of approximate W-disjoint orthogonality of speech in mixtures of various order, and show that even with imperfect W-disjoint orthogonality the histogram can be used to determine the mixing parameters and separate sources. Example demixing results can be found online: http://www.princeton.edu/∼srickard/bss.html

Survey of Sparse and Non-Sparse Methods in Source Separation

Source separation arises in a variety of signal processing applications, ranging from speech processing to medical image analysis. The separation of a superposition of multiple signals is accomplished by taking into account the structure of the mixing process and by making assumptions about the sources. When the information about the mixing process and sources is limited, the problem is called ‘blind’. By assuming that the sources can be represented sparsely in a given basis, recent research has demonstrated that solutions to previously problematic blind source separation problems can be obtained. In some cases, solutions are possible to problems intractable by previous non‐sparse methods. Indeed, sparse methods provide a powerful approach to the separation of linear mixtures of independent data. This paper surveys the recent arrival of sparse blind source separation methods and the previously existing non‐sparse methods, providing insights and appropriate hooks into theliterature along the way.

The DUET Blind Source Separation Algorithm

This chapter presents a tutorial on the DUET Blind Source Separation method which can separate any number of sources using only two mixtures. The method is valid when sources are W-disjoint orthogonal, that is, when the supports of the windowed Fourier transform of the signals in the mixture are disjoint. For anechoic mixtures of attenuated and delayed sources, the method allows one to estimate the mixing parameters by clustering relative attenuation-delay pairs ext- racted from the ratios of the time-frequency representations of the mixtures. The estimates of the mixing parameters are then used to partition the time-frequency representation of one mixture to recover the original sources. The technique is valid even in the case when the number of sources is larger than the number of mixtures. The method is particularly well suited to speech mixtures because the time-frequency representation of speech is sparse and this leads to W-disjoint ort- hogonality. The algorithm is easily coded and a simple Matlab implementation is presented 1 . Additionally in this chapter, two strategies which allow DUET to be applied to situations where the microphones are far apart are presented; this removes a major limitation of the original method.

DOA estimation of many W-disjoint orthogonal sources from two mixtures using DUET

A novel direction of arrival (DOA) technique is presented which constructs estimates of the relative delay mixing parameters associated with each signal by taking the ratio of time-frequency representations of two mixtures. The technique is based on the degenerate unmixing and estimation technique (DUET) (Jourjine et al., Proc. ICASSP 2000, June 5-9, 2000, Istanbul, Turkey). If the sources are W-disjoint orthogonal, meaning that only one signal is active in the time-frequency plane at a given time-frequency, then the ratio only depends on the mixing parameters of one source. The ratio can thus be used to generate estimates of the mixing parameters and these estimates can be clustered to determine both the number of sources present in the mixtures and their associated mixing parameters. The method allows for the estimation of the DOA for many sources using only two receive antennas, whereas traditional techniques require N antennas to estimate N-1 angles of arrival. Simulation results are presented and compared to MUSIC, ESPRIT, and other DOA estimation techniques.

Comparing measures of sparsity

Sparsity is a recurrent theme in machine learning and is used to improve performance of algorithms such as non-negative matrix factorization and the LOST algorithm. Our aim in this paper is to compare several commonly-used sparsity measures according to intuitive attributes that a sparsity measure should have. Sparsity of representations of signals in fields such as blind source separation, compression, sampling and signal analysis has proved not just to be useful but a key factor in the success of algorithms in these areas. Intuitively, a sparse representation is one in which a small number of coefficients contain a large proportion of the energy. In this paper we discuss six properties (robin hood, scaling, rising tide, cloning, bill gates and babies) that we believe a sparsity measure should have. The main contribution of this paper is a table which classifies commonly-used sparsity measures based on whether or not they satisfy these six propositions. Only one of these measures satisfies all six: the Gini index.

Underdetermined blind source separation in echoic environments using DESPRIT

The DUET blind source separation algorithm can demix an arbitrary number of speech signals using anechoic mixtures of the signals. DUET however is limited in that it relies upon source signals which are mixed in an anechoic environment and which are sufficiently sparse such that it is assumed that only one source is active at a given time frequency point. The DUET-ESPRIT (DESPRIT) blind source separation algorithm extends DUET to situations where sparsely echoic mixtures of an arbitrary number of sources overlap in time frequency. This paper outlines the development of the DESPRIT method and demonstrates its properties through various experiments conducted on synthetic and real world mixtures.

Results of the Enumeration of Costas Arrays of Order

This correspondence presents the results of the enumeration of Costas arrays of order 27: all arrays found, except for one, are accounted for by the Golomb and Welch construction methods.