Open access peer-reviewed chapter

Source Separation and DOA Estimation for Underdetermined Auditory Scene

By Nozomu Hamada and Ning Ding

Submitted: April 13th 2012Reviewed: January 30th 2013Published: March 5th 2014

DOI: 10.5772/56013

Downloaded: 1099

1. Introduction

In human-machine communication the separation of a target speech signal and localization of it in noisy environments are very important tasks. [1] For carrying out these tasks recent advanced sensor array signal processing is promising technology. [2] It utilizes the collection of multi-channel acoustic data by an array of microphones for detecting and producing output signals which is much more intelligible and suitable for communication and automatic speech recognition. [3]

BSS problem

Blind source separation (BSS) aims to estimate source signals by only using their mixed signals without any a priori information about mixing process and acoustic circumstances. The cocktail-party problem is one of the typical BSS problems. [1] Basically, the BSS problem can be solved by exploiting intrinsic properties of speech signals. Depending on the inherent properties there have been proposed lots of methods for BSS problems on speech signals. Among them the most widely applied approaches are the following two.

  1. Independent component analysis (ICA)[4]-[8], and

  2. Time-Frequency sparseness of source signals [9]-[14].

The ICA-based separation relies on statistical independence of speech signals in time-domain [5][7] as well as in frequency-domain [6]. In addition, [8] proposed a dynamic recurrent separation system by exploiting the spatial independence of located sources as well as temporal dependence. On the other hand the second approach exploits the sparseness of speech signals in time-frequency (T-F) domain where only small number of T-F components are dominant in representing a speech signal. The T-F sparseness leads the disjoint property of T-F domain components, called W-disjoint orthogonality (WDO) property [11] [12], between speech signals. It means that at most one source dominates at every T-F points, in another word; different speech signals rarely generate the same frequency at the same time.

Though ICA approach performs well even in a reverberant condition, it is difficult to solve the underdetermined case in which the number of sources is greater than the number of sensors. Additionally, the frequency-domain ICA [6] the permutation ambiguity of its solution is a serious problem. It needs to align the separated frequency components that originate from the same source.

The T-F masking method which is the most popular sparseness-based approach is the topic concerned in this chapter. The representative method is known as DUET (Degenerate Unmixing Estimation Technique) [11]. A flow of conventional sparseness-based separation can be summarized as follows.

Sparseness-based T-F masking

Observed signals in T-F domain:

Transform time domain acoustic observations during few seconds to the T-F domain signals by applying short time Fourier transform (STFT) where a sparse representation of speech signal is obtained. [15] Thus the T-F components of a speech signal distribute in T-F domain without overlapping with T-F components of other speech signals.

Features of T-F cells:

As known in auditory scene analysis interaural time differences and level differences are significant spatial features of sources. [1] These localization cues are estimated from the differences in the direction and the distance of speakers. Actually, in microphone array the geometric parameters of sources can be obtained from phase differences and attenuation ratios at the mixture T-F cells.

Clustering T-F cells:

Under the WDO assumption the distribution of feature vectors obtained at all T-F cells makes as many clusters as the number of sources. The essential task of separation therefore turns out to cluster the feature vectors. The preliminary clustering method adopted in [9] - [12] is to make the histogram of features and to find the peaks corresponding the sources. Each T-F cell in the mixed signal is thereby associated with one peak depending on the distance in the cell’s feature space.

Masking T-F cells:

Utilizing the clustering results individual binary masks are applied to the T-F domain spectrogram to detect the components that originate from individual sources.

Inverse transform:

A set of masked T-F components are inversely transformed by STFT and then it provides restored speech signal.

Remarks:

  1. T-F domain sparseness in speech signals is also employed as a separation principle in the context of single channel or monaural signal source separation problem where harmonic structure in spectrogram is crucial for segregation.[16] [17]

  2. Associated with the features of T-F cells conventionally used features are summarized in [13] and the features are evaluated from the separation performance point of view.

  3. Clustering scheme in T-F masking would be crucial for high separation ability. Subsequent studies after DUET-like approaches [11][12], maximum-likelihood (ML) based method for real-time operation [18], k-means algorithm or hierarchical clustering, and EM algorithm [19] have been proposed. The method called MENUET [13] applies k-means algorithm to a vector space consisting of the signal level ratio and the frequency-normalized phase difference with appropriately weighting terms for effective clustering. They solve the optimization problem by adopting an efficient iterative update algorithm. In [14] k-means algorithm is applied clustering spatial features for arbitrary sensor array configuration even with wider sensor distance where spatial aliasing may occur. Their clustering procedure is divided into two steps, the first one of which is applicable to the non-aliasing or lower frequency band and the second one treats the remaining aliasing occurred frequency band.

DOA estimation

Localization of acoustic sources using microphone array system is a significant issue in many practical applications such as hands-free phone, camera control in video conference system, robot audition, and so on. The latter half of this chapter focuses on the Direction-Of-Arrival (DOA) estimation of sources. Since this monograph interests in speech signals, we make no mention of the methods addressed for narrow-band signals, for instance in radar/sonar processing. There have been proposed a large number of DOA estimation methods for broadband signals [20], [21]. Typical array processing approaches are;

  1. Generalized Cross-Correlation (GCC) methods [22]

  2. Subspace approaches using spatial covariance matrix of observed signals [23]

  3. T-F domain sparseness-based approaches [11],[24]-[27]

  4. ICA separation based approaches [28]

The first category of GCC method is to estimate the delay time that maximizes a generalized cross-correlation function between the filtered outputs of the acquired signals at microphones. The phase transform (PHAT) method [22] exploits the fact that the Time-Delay-Of-Arrival (TDOA) information is conveyed in the phase. Although GCC methods are usually performed well and are also computationally efficient for single source case, it does not cope with multiple sources case in which this chapter interests.

The second category is the subspace analysis applying a narrowband signal model. The analysis uses the properties in the spatial covariance matrix of multichannel array observations. The MUSIC-like algorithms are well-known methods for narrowband target signals. For broadband signals such as speech, several frequency-domain approaches have been proposed. The subspace-based approaches for small number of sensors have to overcome two drawbacks, one of which is the limited precision for DOA estimation, and the other is that it is unable to deal with the underdetermined case.

Sparseness-based approaches

The third category of the DOA estimation algorithms is based on sparseness of speech signals and is closely related to the BSS. Source sparseness assumption implies WDO or its weaker condition TIFROM [24]. These conditions are the crucial properties to solve DOA problems for underdetermined multiple sources. The BSS approach associated with these assumptions is a group of T- F masking framework. In DUET-like methods [9]-[14], the delay time or the frequency-normalized ratio of the frequency-domain observations at each T-F point is used to compute the TDOA. An alternative DOA estimation method proposed by Araki et al. [27], in the context of k-means algorithm, estimates DOA as the individual centroid of each cluster of normalized observation vectors corresponding to an individual source. The DEMIX [25] algorithm introduces a statistical model in order to exploit a local confidence measure to detect the regions where robust mixing information is available. The computational cost of DEMIX would be high due to performing the principal component analysis for every local scatter plot of observation vectors at individual T-F points.

For addressing robust cocktail-party speech recognitions the localization cue such as TDOA or spatial direction evaluated at each T-F cell has a central role. As in [29][30], integrating approaches the segregation/localization of sound sources and speech recognition against background interferences are significant CASA (Computational Auditory Scene Analysis) front-ends.

DOA Tracking

Not only estimating but also tracking sound sources draws lots of attentions recently in robot auditory systems. For instance, speaker’s DOA tracking by microphone array mounted on mobile robot is the problem of moving sources and moving sensors.

BSS and DOA Problems:

The underlying BSS and DOA estimation problems addressed in this chapter are listed as follows:

  1. Use of a pair of microphones

  2. Multiple simultaneously uttered speech signals under the assumption that the number of sources is known a priori

  3. Underdetermined cases, where the sources outnumber the sensors

  4. The inter-sensor distance is bounded so as to avoid spatial aliasing (for instance, less than 4 cm spacing for an 8 kHz sampling rate)

While stereophonic sensor is the simplest sensor array, the study of how to improve the separation performance and to obtain accurate DOA by a pair of microphones is meaningful because any complex array configuration can be considered as an integration of these.

The rest of this chapter is organized as follows. In section 2, problems of underlying BSS and DOA estimation are described in detail. The proposed BSS method based on a frame-wise scheme is introduced in section 3. Section 4 describes a DOA estimation algorithm by using T-F cell selection and the kernel density estimator. The last section concludes this chapter.

2. Problem descriptions

2.1. Observation model

Source mixing models in time domain and its T–F domain description are described as follows. All discrete time signals are sampled version of analog signals with sampling frequency fS. Suppose N source signals s1(t),s2(t),,sN(t)are mixed by time-invariant convolution and the observed signals x1(t),x2(t),,xM(t)at M sensors with omni-directive characteristic are described as:

xm(t)=i=1Nτhmi(τ)si(tτ),E1

where hmi(τ)represents the impulse response from i-th source to m-th sensor. Observed signals xm(t)(m=1~M) are converted into T–F domain signals Xm[k,l]by using L-point windowed STFT as written by

Xm[k,l]=r=L/2L/21xm(r+kS)win(r)ej2πlLr,k=0K,l=0L2E2

where r is dummy variable in convolution sum operation, win(r) is a window and S is the window shift length. Here, we apply half window size overlapping transformation, namely S = L/2 in (2). Transformed T–F mixture model of Eq.(1) can be described by the instantaneous mixtures at each time frame index k and frequency bin l.

Xm[k,l]=i=1Nmi[l]Si[k,l]E3

where Hmi[l]is the frequency response (DFT) of hmi(t), Si[k,l]is the windowed STFT representation of i-th source signal si(t), and the point [k,l]is called “T-F cell” in this chapter. Assuming an anechoic mixing, the source signals which we want to recover are alternatively redefined as the observed signals at the first mixture x1[k,l]. In this case, the following mixing models in the T–F domain are henceforth considered without loss of generality.

X1[k,l]=i=1NSi[k,l],                                                aXm[k,l]=i=1NHmi[l]Si[k,l]m=2M     bE4

where Si[k,l]and Hmi[l]are different from Si[k,l]and mi[l]in (3), Si[k,l]is the i-th source signal observed at the first sensor (m=1), and Hmi[l]eventually represents the DFT domain operation of the transfer function with relative attention and delay between m-th and the first sensors.

From then on, consider the mixture of two sources S1[k,l]and S2[k,l]which are received at a pair of microphones. Their mixture system (4a) and (4b) can thus be expressed as

[X1[k,l]X2[k,l]]=[1,1H21[l],H22[l]][S1[k,l]S2[k,l]]E5

2.2. Basic assumptions

As stated in Section 1, the WDO is commonly supposed in sparseness-based separation approaches. At first, we denote the T-F domain Ωon which S1[k,l]and S2[k,l]are defined

Ω:={[k,l],k=0K,lB}E6

where B:=[l1,L/2]is the frequency band after deleting lower frequency components which do not exist in actual speech signals, and l1=f1L/fsmeans the Gauss floor function, and f1is the analog lowest frequency of speech components such as 80Hz in later experiments.

Next, define the T-F supports Ωi(i=1,2)of Si[k,l](i=1,2)by

Ωi:={[k,l]||Si[k,l]|>ε}i=1,2E7

where ε(>0)is a sufficiently small value. Although, in theory, the support of Si[k,l](i=1,2)is defined by the condition |Si[k,l]|0, Eq. (7) gives a set of components of actual signals except noise-like ones satisfying |Si[k,l]|<ε.

We may consequently express the WDO assumption between two source signals s1(t)and s2(t)by the disjoint condition

Ω1Ω2=ϕ(emptyset)E8

This can equivalently be represented as follow.

S1[k,l]S2[k,l]=0      at any [k,l] E9

The verification of above WDO condition for actual speech signals is performed in Fig. 1 where (a) and (b) show spectrograms of two speech signals in the T-F domain, and (c) shows their multiplication in which we see rarely overlapping between two spectrograms.

Figure 1.

WDO property of two real speech signals

Obviously, the supports of X1[k,l]and X2[k,l]are coincident and it, denoted by ΩX, can be given as

ΩX=Ω1Ω2E10

In addition the following null component domain, denoted by ΩN, is also introduced as

ΩN=Ω¯X=Ω1Ω2¯X¯:complementary set of XE11

The WDO condition (8) accordingly derives that the T–F domain representation of the mixed signal X1[k,l], given by Eq.(5), can be decomposed into the following three parts with no overlapping in Ω.

X1[k,l]={S1[k,l][k,l]Ω1S2[k,l]0[k,l]Ω2[k,l]ΩNE12

2.3. Source separation

Under the WDO assumption expressed in (12), the binary masking in the T–F domain is performed as follow:

Clustering the T-F cells in the support ΩXof the mixture X1[k,l]into two sub-regions Ω1and Ω2, the separated source estimates in T-F domain, S^1[k,l]and S^2[k,l], are obtained by applying the masks

Mi[k,l]={1[k,l]Ωi0otherwise.(i=1,2)E13

on X1[k,l]as follows.

S^i[k,l]=Mi[k,l]X1[k,l](i=1,2)E14

Clustering features

The separation task is to classify T–F cells composing the support ΩXof X1[k,l]into either Ω1or Ω2. A pair of X1[k,l]and X2[k,l]is used to characterize a T-F cell [k,l]at which spatial features are introduced, and the clustering process is performed in the estimated feature space.

Effective features must be the signal level or attenuation ratio defined by

α[k,l]:=|X1[k,l]X2[k,l]|E15

and the arrival time difference defined by the frequency-normalized phase difference (PD) between X1[k,l]and X2[k,l]as

δ[k,l]:=L2πfslϕ[k,l]E16

where ϕ[k, l] is the PD as defined by

ϕ[k,l]=X1[k,l]X2[k,l]E17

Other features used for characterizing T-F cells are listed in [13]as well as the attenuation ratio modifications. It is noted that the attenuation ratio would not give distinctive difference for short distance microphone array. In our experimental setup, for example, the distance between microphones is 4cm in order to avoid spatial aliasing at 8kHz sampling rate.

Clustering scheme

For given features at T-F cells in ΩX, clustering of these is the next step. In DUET where a pair of microphones is used, the two dimensional histogram of feature vectors {α[k,l],δ[k,l]}Twithin a time interval, such as for several seconds, is generated and the clustering is performed by finding the maximum peaks which are corresponding to sources. When the attenuation feature is omitted the clustering problem is solely performed based on time delay histogram distribution. The dimension of feature space will be higher for array configuration with many microphones than two. For these cases more sophisticated clustering scheme such as k-means algorithm or EM algorithm [19] should be adopted.

Inverse STFT

The final stage of the separation process is to obtain time domain separated signals s^i(t)(i=1,2)by applying the inverse STFT.

3. Sound source separation

3.1. Phase–difference vs. frequency data

As a T-F cell’s feature depending on the spatial location difference of sources, our strategies exploit a frame-wise, namely, a time sequence of phase difference of observations versus frequency (PD-F) distribution. In a k-th frame, the point plot of the PD-F is defined as a collection of two-dimensional vectors at k-th frame pk(l)as

pk(l):={l,ϕ[k,l]}T,lBk[1,K]E18

An example of PD-F in (l,ϕ)-plane and its time sequence for the mixture of two speech signals are shown in Fig.2 (a) and (b) respectively.

Figure 2.

PD-F and time sequence of PD-F (Blue and red points respectively corrrespond invividual source components)

The relationship between the gradient β of a vector in PD-F plane defined in Eq.(18) and the source direction θ is: [33]

β=(2πL)fsdcsinθ E19

where d is the distance between the sensors, c is the sound velocity, and θis the direction of source. Here θ=0corresponds to the broadside direction and the term (d/c)sinθrepresents the wave arriving delay between microphones. For example, the dot distribution in Fig.2 (a) concentrates along two lines corresponding to two source directions. By determining the gradients of these lines two directions of sources are estimated from the relationship of (19).

The conventionally utilized features associating with delay time at each T-F cell can be estimated from the frequency normalization of PD-F dot corresponding to individual T-F cells. Unlike the conventional delay-like features PD-F dots keep a linear dot distribution on the plane and it is effectively utilized in both following source separation and direction finding methods.

3.2. Frame categorization

Fig. 3(a) shows two simultaneously uttered speech signals. In the figure four frame time points k1 - k4 indicated by the red rectangular parts are shown as the following four types of source signal activity states:

Frame k=k1; No source signal is active (Non Source Active:NSA)

Frame k=k2; Only the first source is active (Single Source SSA)

Frame k=k3; Only the second source is active (Single Source SSA)

Frame k=k4; Both sources are active (Double Source Active:DSA)

Here we may define three sets of time-frame indeces as follows:

KNSA:={k|NSA frames},KSSA:={k|SSA frames},KDSA:={k|DSA frames}

The whole set of time-frames, denoted by K:={1,.....,K}, is categorized into three sets with no overlapping.

K=KNSAKSSAKDSAE20

In addition, we define the following Sourse Active(SA) frame index set.

KSA=KSSAKDSA E21

Figure 3.

Frame categorization (NSA, SSA, DSA)

Above frame categorization suggests the source separation algorithm consisting of the following two parts:

  • Assign each T-F component at SSA frame to either source by identifying the direction.

  • Apply separation algorithm solely to DSA frames

The detail of these will be described in the next section.

3.3. Source separation algorithm

Outline of the method

The outline of the separation method using PD-F plot is shown in Fig.4 and summarized.

Figure 4.

Flow of source separation method

Step1: Discriminate SA from NSA

The following average power at a frame is employed to check the presence of speech signal at the frame.

E(k):=1L/2l1+1lB|X1[k,l]|2E22

Here, the threshold operation is valid for basic voice activity detection as follow.

KSA={k|E(k)>ThSA}E23

where ThSA is determined by a pre-experiment of noise level estimate during no utterance. In later experiments, we applied the following formula.

ThSA=E0+2σEE24

where E0 is the average noise power estimate and σEis the standard deviation estimate given by respectively.

E0:=1|kNSA|kkNSA|X1[k,l]|2E25
σE:=1|kNSA|kkNSA(E(k)E0)2E26

Step 2-1: Classify SA into SSA and DSA

At each kKSAPCA is applied to the set of vectors pk(l)by computing the following 2 × 2 covariance matrix.

Rk:=1L/2l1+1lBpk(l)pkΤ(l)=[R11(k)R12(k)R21(k)R22(k)].E27

Denoting the eigenvalues of Rkby λ1(k) and λ2(k) (assume λ1(k) ≥ λ2(k)), the ratio of the principal eigenvalues defined by

r(k):=λ2(k)λ1(k). E28

is introduced to discriminate the SSA frames from the DSA. As shown in Fig.3 (c), (d), PD-F vector distribution at a SSA frame tends to concentrate around the first principal axis. This observation leads to the following discrimination of SSA from DSA frames and the estimation of the source directions.

The following criterion is applied to detect SSA frames.

KSSA={k|r(k)<ThSSA}E29

where ThSSA is determined experimentally.

Step 2-2: DOA estimation and SSA identification

Define the normalized eigenvector of the first principal eigenvalue as

e1(k):=[cosβ(k)sinβ(k)] E30

where β(k) is the gradient of the principal axes at k-th frame. The histogram of the set

{β(k),kKSSA}E31

will have two peaks which are corresponding two source directions θ1and θ2caluculated by Eq. (19). By clustering the set of θinto two groups according to the distance from θ1 and θ2, each SSA frame in KSSAis classified into each one of the sources from the direction θ1 and θ2.

Double Source Active (DSA)

For given set of DSA frames KDSA, the clustering of the vectors pk(l),lBinto two sets is the problem. Before describing this separation algorithm, three frequency bands, denoted by Bhigh, Blow, and Bmid, are introduced to use in the following separation algorithm.

Frequency Bands

The following three frequency bands are defined respectively.

Bhigh:={l|l2<l<L/2},Blow:={l|l1<l<l2},Bmid:={l|l2<l<l3}

where li=fiL/fs,(i=2,3), f2is set 400Hz, and f3 is set 1kHz in later experiments.

The idea of source separation at DSA frames utilizing these bands is divided into two parts according to above frequency bands.

  1. The first scheme, called initial separation, is applied to the T-F cells in Bhighbased on the directions of sources which have been estimated at the SSA frames previously.

  2. The clustering in Blowis performed utilizing a harmonic structure relationship between the spectral components in Blowand that of Bmid. The harmonic structure in Bmidcan be obtained by the initial separation results in Bhigh.

  1. Initial separation

Denote the source directions estimated in SSA frames by θ1 and θ2, and their corresponding gradients in PD-F plane are β1 and β2 as defined in Eq.(31). The points on these two lines can be expressed as

ϕ[k,l]=βil(i=1,2)E32

At kKDSA, the nearest neighbor rule gives the binary mask M˜i[k,l]in Bhigh which is defined as

M˜i[k,l]={1,ifi=argminc|ϕ[k,l]βcl|,lBhigh0,otherwise.E33

As a result, the separated individual signals S˜i[k,l](i = 1, 2) are represented by

S˜i[k,l]=M˜i[k,l]X1[k,l],lBhighE34
  1. Separation in Blow

Local maximum points in Bmid

The final task for separation process is to generate individual mask applied to Blow. In this final separation process, the observed amplitude spectrum given by |X1[k, l]| with lBlow is compared with the initially separated spectra S˜1[k,l]and S˜2[k,l]with lBmid in terms of harmonic relationships. At first, with the help of local maximum frequencies of |S˜i[k,l]|, harmonic structure in Bmid is estimated for each separation spectra. We denote the obtained local maximum frequencies of |S˜i[k,l]|are bi1(k), bi2(k), ・ ・ ・, and the number of local maxima in Bmid is qi(k).

Harmonics estimation

The distance of adjacent harmonics Δdi(k) is defined as

Δdi(k)=bi2(k)bi1(k),qi(k)>2  E35

When qi(k) = 0 or 1, we regard that there is no harmonic in the frame k. The estimated harmonics in low frequency band gin(k) is

gin(k)=bl1(k)Δdi(k)n,E36

where n=1,2,3,,gin(k)Blow, and gin(k)means the harmonic structure of source i at frame k.

Massk generation

We assume that the bandwidth of each harmonics is the same, and use 5 adjacent cells as bandwidth in T–F domain. The mask in Blow is defined

M¯i[k,l]={1,ifgin(k)2<l<gin(k)+2,and                        qi(k)2,lBlow,n=1,2,3,0,otherwise.E37

The integrated mask combining Eq. (33) and Eq. (37) is represented by

Mi[k,l]=M˜i[k,l]+M¯i[k,l].E38

Finally, the separated signals are obtained as shown in Eq.(14).

3.3. Experiments

Experimental condition

Some real life experiments are performed in a conference room to evaluate the separation methods. Fig.5(a),(b) show the experimental environments and the setup. The experimental parameters are show in Tab.1. One source was placed at the broadside (θ=0) and the location of the other source is varied from 0 to 80 at intervals of every 10.

Figure 5.

Experiments

Figure 6.

Experimental results (SIR)

Fig. 6 shows the average signal-to-interference ratio (SIR) improvement brought by the proposed and the conventional DUET method. The SIR improvement at the first sensor is defined as follows.

                SIRi improvement=Output SIRiInput SIRi    E39

Where

 Input SIRi=10log10si(t)sj(t) ,   Output SIRi=10log10yii(t)yij(t)  

The proposed frame-wise PD-F approach exceeds the conventional method in terms of SIR improvement. The average improvement in our experiments is 6.22dB over the DUET. The most significant contribution in SIR improvement is made by the separation process in DSA frame which is 4.28dB. [31]

Source signal duration5s speech signals
Sampling Frequency8 kHz
Sound Velocity340 m/s
WindowHamming
STFT Frame Length1024 sample
Frame Overlap512 sample

Table 1.

Experimental Parameters

4. DOA estimation

The DOA estimation method discussed in this section is based on the following three novel approaches.

  1. Inspired by the ideas of TIme-Frequency Ratio Of Mixtures (TIFROM)-like assumptions, a novel reliability index is introduced. The selected cells with higher reliability are solely utilized for DOA estimation.

  2. A statistical error propagation model relating PD-F and the consequent DOA is introduced. The model leads to a probability density function (PDF) of the DOA, and hence the DOA estimation problem is reduced to finding the most probable points of the PDF.

  3. Source directions are determined using the kernel density estimator by utilizing the proposed bandwidth control strategy.

DOA information

Under the assumption of anechoic mixing with no-attenuation model and WDO in Eq. (5), the ratio between two observations Xm[k, l] (m=1,2) is represented by

X2[k,l]X1[k,l]=H2n[l]H1n[l]=exp[j2πfslL×dcsinθ],E40

where θis the direction of source which is dominant at [k,l]. The phase difference (PD) ϕ[k,l] between two observations Xm[k, l] (m=1,2) defined by Eq. (17) is related to the angle θas follows.

ϕ[k,l]=2πfsldLcsinθ=ΔωTlsinθ, E41

where T =d/c is the maximum delay time between sensors, and Δω=2πfS/Lis the unit frequency width in L-point STFT. From Eqs. (16) and (41), the TDOA normalized by T, denoted by τ[k,l], can be represented as follows.

τ[k,l]=sinθ=ϕ[k,l]TΔωlE42

4.1. Reliable T–F cell selection

As stated in 2.2, the following selection processes are applied only to the T-F cells in the support ΩXof X1[k,l]as in 2.2. This eventually reduces the computation time by eliminating noise-like T-F components.

Since the PD estimation by (17) is subjected to unavoidable error, the success of DOA estimation is generally expected if reliable PD data are selected to use and outliers are eliminated. Likewise in [24], the following assumption is employed. When a source is dominant in a set of cells, all delays in it will take almost the same value; hence, the delay (42) and obviously the PD data (17) in this set are expected to be reliable. Conventionally, the confidence measure is obtained from the results of applying the principal component analysis to a set of steering vectors in individual horizontal and vertical T-F regions. Unlike this approach, the normalized delays τ[k,l]given by Eq.(42) are used to evaluate the attribute consistency of the T-F cells. According to the above assumption, two types of T-F regions around a cell [k, l] are considered: a temporal neighborhood Γt[k, l] and a frequency neighborhood Γf [k, l],

Γt[k,l]:={[k+y,l]||y|Y},Γf[k,l]:={[k,l+z]||z|Z},E43

where integers Y and Z determine the numbers of cells in these regions, as denoted by |Γt[k, l]| := 2Y + 1 and |Γf [k, l]| := 2Z + 1.

For each Γt[k, l] and Γf [k, l], the standard deviations of the normalized delays σΓt [k, l] and σΓf [k, l] are calculated by

σΓ[k,l]=1|Γ|[p,q]Γ(δ[p,q]μΓ[k,l])2E44
μΓ[k,l]=1|Γ|[p,q]Γδ[p,q],Γ=Γt,Γf.E45

Now, the reliability index η[k, l] is calculated by

η[k,l]=exp{min(σΓt[k,l],σΓf[k,l])}E46

where η[k, l] is a normalized index satisfying 0 < η ≤ 1. When at least σΓt [k, l] or σΓf [k, l] at [k, l] is sufficiently small, η[k, l] approaches unity, thereby the corresponding delay value δ[k, l] is considered to be reliable. We observed the tendency that the PD error decreases as the reliability index increases. Then, the cell group is selected with reliability index η[k, l] > ηth for subsequent DOA estimation. In this paper, ηth is set to 0.96. The reason for using this value and related remarks are given in later.

For each selected reliable T-F cell, the direction θis computed using Eq.(41). Here the set of computed directions is denoted as follows:

{θi[li]|i=1,2,...,I}, E47

where i is the numbering integer of the selected cells, I is the total number of data, and li is the frequency bin at which the i-th cell is located.

DOA error distribution model

Consider a T-F cell at which the n-th source dominates and is located in the unknown direction θn. From Eq. (41), the theoretical PD at the cell is given by

ϕn[l]=ΔωTlsinθn=Bnl, E48

where Bn = ΔωT sinθn. The frame index k is omitted because k is not essential in this section. In the l-th frequency bin, the observed ϕn[l] is distributed around its mean value Bnl,

ϕn[l]=BnI+Δϕ[l],E49

where Δϕ[l] is a random variable representing the PD estimation error. Then, assume that the random variable Δϕ[l] is an independent identical Gaussian distribution with zero mean and constant variance σϕ2, that is, N(o,σϕ2). The constant variance means that Δϕ[l] is independent of the frequency bin l; this assumption is represented as follows:

Δϕ[l]N(0,σ2ϕ).E50

Fig. 7 (a) illustrates Gaussian error distribution at l-th frequency bin in PD-F plane in two-source case. The Gaussian distribution assumption is motivated from the simplicity of theoretical manipulation. From these error distribution model the problem is to estimate the probability distribution of the direction θ as shown in Fig.7(b).

Now, the following proposition can be proved.

Figure 7.

PD error distribution and Kernel density estimation

Proposition: If the random variable Δϕ[l] is given by (50) and σϕ is sufficiently small,

the PDF of θn[l]is given by

θn[I]N(θn,σθn2[l]), E51
σθn[l]=1TΔωlcosθnσϕ.E52

This proposition can be proved by the linearized incremental analysis between ϕ[l] and θ[l]. The DOA error distribution model is shown in Fig. 8.

Figure 8.

PD error and DOA estimation error distributions

4.2. DOA estimation using kernel density estimator

The kernel density estimation algorithm known as Parzen window in machine learning [32] is useful for statistical estimation even for a multiple-source problem. The algorithm provides an estimate PDF of θ[l] by using the observed samples (47). The maximum PDF point or the mode of the PDF can be considered as the optimal estimate of θn in the sense of the most probable value. The kernel density estimator approach yields an approximate estimation of the PDF of θ[l].

It is necessary to generalize the theoretical investigation noted above multisource and multi-frequency cases. The theoretical PDF formulation of θ in the case of multiple sources should be a Gaussian mixture with the same number of local modes (local peaks), each of which corresponds to an individual source. For the selected reliable data in Eq. (47), the kernel density estimator is applied to estimate the multi-modal PDF as follows:

p^(θ)=1Ii=1I1ε[li]K(θθi[li]ε[li]),E53

where K(θ) is a kernel function, for which a Gaussian function is adopted in this study. ε[l]is the bandwidth of the kernel. The idea behind applying the kernel density estimator is to reflect the theoretical result represented by the above proposition for the determination of the bandwidth. Since the variance of θ[l] depends on l and θn as indicated in Eq. (52), the bandwidth is determined as the form of

ε[Ii]=1TΔωlicosθi[Ii]. E54

where is the control parameter and the observed θ[li] is substituted in place of a real unknown θn in Eq. (52). Accordingly, the dependence of the bandwidth on θn is indirectly controlled. The control parameter is predetermined experimentally. Fig. 9 shows three examples of estimated PDFs for a two-source case with different . Finally, by finding the same number of local modes (peaks) as the number of pre-assigned source numbers, the source directions are estimated.

4.3. Experiments

Some experiments were conducted by the same setup and parameters as shown in Tab. 1. The first experiment is the case of two sources one of which is placed at the broad side (near 0 degree) as shown in Fig.10 (a). The results are shown in Fig.10 (b) and (c). While the proposed method gives a non-biased estimation, the estimates of the conventional method [27] tend to be biased for the cases of non-symmetric source positions with respect to the broadside. The second experiments for underdetermined case of three sources were performed. In this case three sources were symmetrically located at the closer locations { -23, 4, 23 degrees} and far apart locations{ -42, 4, 42 degrees}. Fig.11 (a) and (b) show the results of the conventional method [27] and the proposed. In the “far apart” case both methods can estimate the source directions well. However, for the “closer” case, the proposed method provides less biased estimates than [27]. From the additional experimental results with diffuse noise presented in [33] and [34] it is proved the proposed cell selection method provides noise robust estimation better than the conventional.

Figure 9.

Estimated PDF and ℏ

Figure 10.

DOA estimation results for two sources

Figure 11.

DOA estimation results for three sources

5. Conclusions

This monograph summarizes speech segregation and speaker’s direction estimation methods which are based on sparseness of T-F components of speech signals. Throughout the discussion we are interested in underdetermined source-sensor conditions. At first recent progresses on BSS and DOA estimation algorithms associated with T-F sparse representation are reviewed. Then we focus on presenting an author’s solution of BSS problems exploiting a series of phase difference versus frequency data. In the algorithm time frame classification concerning source active states is performed, and actual separation procedure is solely applied to the mixing frames.

The latter half of this chapter treats DOA estimation algorithm in a pair of microphones.

The basic error propagating mechanism is introduced and then the kernel density estimator is applied. The method provides a robust and non-biased DOA estimation and it develops theory for arbitrary microphone array configuration. [35]

One of recent human machine speech communication research on segregation and localization is associated with robot auditory system where the tracking of moving sources and sensors have to be considered.[36] For coping with these cases the particle filter and adaptive array processing have been attractive, and further efforts will be made.

Acknowledgments

The authors would like to appreciate Professor Wlodzimierz Kasprzak of Warsaw University of Technology for his valuable suggestions, and all members of speech signal processing group of Hamada Laboratory in Keio University for their great help.

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Nozomu Hamada and Ning Ding (March 5th 2014). Source Separation and DOA Estimation for Underdetermined Auditory Scene, Soundscape Semiotics - Localization and Categorization, Herve Glotin, IntechOpen, DOI: 10.5772/56013. Available from:

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