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8 The DUET Blind Source SeparationAlgorithm

Scott Rickard

University College DublinBelfield, Dublin 4, IrelandE-mail: scott.rickard@ucd.ie

Abstract. This chapter presents a tutorial on the DUET Blind Source Separationmethod which can separate any number of sources using only two mixtures. Themethod is valid when sources are W-disjoint orthogonal, that is, when the supportsof the windowed Fourier transform of the signals in the mixture are disjoint. Foranechoic mixtures of attenuated and delayed sources, the method allows one toestimate the mixing parameters by clustering relative attenuation-delay pairs ext-racted from the ratios of the timefrequency representations of the mixtures. Theestimates of the mixing parameters are then used to partition the timefrequencyrepresentation of one mixture to recover the original sources. The technique isvalid even in the case when the number of sources is larger than the number ofmixtures. The method is particularly well suited to speech mixtures because thetimefrequency representation of speech is sparse and this leads to W-disjoint ort-hogonality. The algorithm is easily coded and a simple Matlab implementationis presented1. Additionally in this chapter, two strategies which allow DUET tobe applied to situations where the microphones are far apart are presented; thisremoves a major limitation of the original method.

8.1 Introduction

In the field of blind source separation (BSS), assumptions on the statisti-cal properties of the sources usually provide a basis for the demixing algo-rithm [1]. Some common assumptions are that the sources are statisticallyindependent [2, 3], are statistically orthogonal [4], are nonstationary [5], orcan be generated by finite dimensional model spaces [6]. The independenceand orthogonality assumptions can be verified experimentally for speech sig-nals. Some of these methods work well for instantaneous demixing, but failif propagation delays are present. Additionally, many algorithms are compu-tationally intensive as they require the estimation of higher-order statisticalmoments or the optimization of a nonlinear cost function.

One area of research in blind source separation that is particularly chal-lenging is when there are more sources than mixtures. We refer to such a case

This material is based upon work supported by the Science Foundation Irelandunder the PIYRA Programme.

1The author is happy to provide the Matlab code implementation of DUETpresented here.

217S. Makino et al. (eds.), Blind Speech Separation, 217241. 2007 Springer.

218 Scott Rickard

as degenerate. Degenerate blind source separation poses a challenge becausethe mixing matrix is not invertible. Thus the traditional method of demixingby estimating the inverse mixing matrix does not work. As a result, mostBSS research has focussed on the square or overdetermined (nondegenerate)case.

Despite the difficulties, there are several approaches for dealing with deg-enerate mixtures. For example, [7] estimates an arbitrary number of sourcesfrom a single mixture by modeling the signals as autoregressive processes.However, this is achieved at a price of approximating signals by autoregres-sive stochastic processes, which can be too restrictive. Another example ofdegenerate separation uses higher order statistics to demix three sources fromtwo mixtures [8]. This approach is not feasible however for a large numberof sources since the use of higher order statistics of mixtures leads to an ex-plosion in computational complexity. Similar in spirit to DUET, van Hulleemployed a clustering method for relative amplitude parameter estimationand degenerate demixing [9]. The assumptions used by van Hulle were thatonly one signal at a given time is nonzero and that mixing is instantaneous,that is, there is only a relative amplitude mixing parameter associated witheach source. In real world acoustic environments, these assumptions are notvalid.

DUET, the Degenerate Unmixing Estimation Technique, solves the deg-enerate demixing problem in an efficient and robust manner. The underlyingprinciple behind DUET can be summarized in one sentence:

It is possible to blindly separate an arbitrary number of sources givenjust two anechoic mixtures provided the timefrequency representa-tions of the sources do not overlap too much, which is true for speech.

The way that DUET separates degenerate mixtures is by partitioning thetimefrequency representation of one of the mixtures. In other words, DUETassumes the sources are already separate in that, in the timefrequencyplane, the sources are disjoint. The demixing process is then simply a parti-tioning of the timefrequency plane. Although the assumption of disjointnessmay seem unreasonable for simultaneous speech, it is approximately true.By approximately, we mean that the timefrequency points which containsignificant contributions to the average energy of the mixture are very likelyto be dominated by a contribution from only one source. Stated another way,two people rarely excite the same frequency at the same time.

This chapter has the following structure. In Sect. 8.2 we discuss the ass-umptions of anechoic mixing, W-disjoint orthogonality, local stationarity,closely spaced microphones, and different source spatial signatures which leadto the main observation. In Sect. 8.3 we describe the construction of the 2Dweighted histogram which is the key component of the mixing parameterestimation in DUET and we describe the DUET algorithm. In Sect. 8.4

8 The DUET Blind Source Separation Algorithm 219

we propose two possible extensions to DUET which eliminate the requirementthat the microphones be close together. In Sect. 8.5 we discuss a proposedmeasure of disjointness. After the conclusions in Sect. 8.6, we provide theMatlab utility functions used in the earlier sections.

8.2 Assumptions

8.2.1 Anechoic Mixing

Consider the mixtures of N source signals, sj(t), j = 1, . . . , N , being receivedat a pair of microphones where only the direct path is present. In this case,without loss of generality, we can absorb the attenuation and delay parame-ters of the first mixture, x1(t), into the definition of the sources. The twoanechoic mixtures can thus be expressed as,

x1(t) =N

j=1

sj(t), (8.1)

x2(t) =N

j=1

ajsj(t j), (8.2)

where N is the number of sources, j is the arrival delay between the sen-sors, and aj is a relative attenuation factor corresponding to the ratio of theattenuations of the paths between sources and sensors. We use to denotethe maximal possible delay between sensors, and thus, |j | ,j. The ane-choic mixing model is not realistic in that it does not represent echoes, thatis, multiple paths from each source to each mixture. However, in spite of thislimitation, the DUET method, which is based on the anechoic model, hasproven to be quite robust even when applied to echoic mixtures.

8.2.2 W-Disjoint Orthogonality

We call two functions sj(t) and sk(t) W-disjoint orthogonal if, for a givenwindowing function W (t), the supports of the windowed Fourier transformsof sj(t) and sk(t) are disjoint. The windowed Fourier transform of sj(t) isdefined,

sj(, ) := FW [sj ](, ) :=12

W (t )sj(t)eitdt. (8.3)

The W-disjoint orthogonality assumption can be stated concisely,

220 Scott Rickard

sj(, )sk(, ) = 0, , , j = k. (8.4)

This assumption is the mathematical idealization of the condition that it islikely that every timefrequency point in the mixture with significant energyis dominated by the contribution of one source. Note that, if W (t) 1,sj(, ) becomes the Fourier transform of sj(t), which we will denote sj().In this case, W-disjoint orthogonality can be expressed,

sj()sk() = 0,j = k,, (8.5)

which we call disjoint orthogonality.W-disjoint orthogonality is crucial to DUET because it allows for the

separation of a mixture into its component sources using a binary mask.Consider the mask which is the indicator function for the support of sj ,

Mj(, ) :={

1 sj(, ) = 00 otherwise. (8.6)

Mj separates sj from the mixture via

sj(, ) = Mj(, )x1(, ),, . (8.7)

So if we could determine the masks which are the indicator functions for eachsource, we can separate the sources by partitioning. The question is: how dowe determine the masks? As we will see shortly, the answer is we label eachtimefrequency point with the delay and attenuation differences that explainthe timefrequency magnitude and phase between the two mixtures, andthese delay-attenuation pairs cluster into groups, one group for each source.

8.2.3 Local Stationarity

A well-known Fourier transform pair is:

sj(t ) ei sj(). (8.8)

We can state this using the notation of (8.3) as,

FW [sj( )](, ) = eiFW [sj()](, ), (8.9)

when W (t) 1. The above equation is not necessarily true, however, whenW (t) is a windowing function. For example, if the windowing function werea Hamming window of length 40 ms, there is no reason to believe that two40 ms windows of speech separated by, say, several seconds are related by aphase shift. However, for shifts which are small relative to the window size,(8.9) will hold even if W (t) has finite support. This can be thought of as a

8 The DUET Blind Source Separation Algorithm 221

form of a narrowband assumption in array processing [10], but this label isperhaps misleading in that speech is not narrowband, and local stationarityseems a more appropriate moniker. What is necessary for DUET is that (8.9)holds for all , || , even when W (t) has finite support, where isthe maximum time difference possible in the mixing model (the microphoneseparation divided by the speed of signal propagation). We formally state thelocal stationarity assumption as,

FW [s