Blind source separation based on high-resolution time–frequency distributions q Jing Guo a,b,⇑ , Xiaoping Zeng a , Zhishun She c a College of Communication Engineering, Chongqing University, Chongqing 400044, China b School of Electronic and Information Engineering, Southwest University, Chongqing 400715, China c Engineering Department, Glyndwr University, University of Wales, LL11 2AW, UK article info Article history: Received 23 October 2010 Received in revised form 10 December 2011 Accepted 12 December 2011 Available online 2 January 2012 abstract Blind source separation (BSS) based on time–frequency distributions (TFDs) exploits the underlying diagonal or off-diagonal structure of TFD matrices to separate the source sig- nals. In this paper, we propose a new signal-independent kernel which is defined in both the time–lag and the Doppler-lag domain and satisfies most of the desirable properties of a TFD. The main objective of this research is to achieve the high resolution and the max- imum cross-term reduction with the preferable diagonal or off-diagonal structure of TFD matrices in BSS applications. Moreover, a BSS approach is developed which includes first whitening mixed signals, then constructing a set of TFD matrices using the proposed TFD and the Hough transform, finally a joint diagonalization of a combined set of TFD matrices to estimate the mixing matrix and the source signals. By use of the techniques proposed in this paper, the improved performance of BSS of nonstationary signals has been achieved. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction The aim of blind source separation (BSS) is to recover the sources form the observations only [1]. Different methods have been developed for BSS based on cyclostationary, second-order and/or high-order statistics of the source signals, linear and quadratic time–frequency (t–f) transforms. In many applications such as radar, sonar and acoustic localization, the signals of interest are known to be nonstationary in nature. For these signals, BSS based on time–frequency distributions (TFDs) pro- vides an improved performance over other methods, when dealing with signals that are localized in the t–f domain. Among the existing TFDs-BSS algorithms [2–10], typically, the TFD matrices are constructed from the auto-terms and cross-terms of the observed signals and used for source diagonalization and antidiagonalization, or a combination of both techniques, depending on the t–f point selections and the structure of the source TFD matrices. These existing methods, hence, allow the separation of source signals with the identical spectral shape but with the different t–f localizations. How- ever, for multicomponent signals, the inherent bilinear structure of Cohen’s class, for example, the Wigner-Ville Distribution (WVD) and/or its expansions causes the interfering cross-terms which often lead to misinterpretation of the signals. Several different approaches [11–15] have been proposed to eliminate the cross-terms interference based on time–fre- quency ratio of mixtures (TIFROM). These approaches are inspired by a method that requires single source to occur alone in a tiny t–f zone, i.e. they set very limited constraints on the source sparsity and overlapping. TIFROM and its expansions rely on identifying the columns of the mixing matrix by first finding the t–f zones where only one source exists and then independently estimating the scale coefficients and the time shifts in these single-source zones using ratios of mixtures. TI- 0045-7906/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compeleceng.2011.12.002 q Reviews processed and proposed for publication to Editor-in-Chief by Guest Editor Dr. Yi Wan. ⇑ Corresponding author at: School of Electronic and Information Engineering, Southwest University, Chongqing 400715, China. Tel.: +86 23 68117847. E-mail address: [email protected](J. Guo). Computers and Electrical Engineering 38 (2012) 175–184 Contents lists available at SciVerse ScienceDirect Computers and Electrical Engineering journal homepage: www.elsevier.com/locate/compeleceng
Transcript
Computers and Electrical Engineering 38 (2012) 175–184
Contents lists available at SciVerse ScienceDirect
Blind source separation based on high-resolution time–frequencydistributions q
Jing Guo a,b,⇑, Xiaoping Zeng a, Zhishun She c
a College of Communication Engineering, Chongqing University, Chongqing 400044, Chinab School of Electronic and Information Engineering, Southwest University, Chongqing 400715, Chinac Engineering Department, Glyndwr University, University of Wales, LL11 2AW, UK
a r t i c l e i n f o a b s t r a c t
Article history:Received 23 October 2010Received in revised form 10 December 2011Accepted 12 December 2011Available online 2 January 2012
0045-7906/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.compeleceng.2011.12.002
q Reviews processed and proposed for publication⇑ Corresponding author at: School of Electronic a
Blind source separation (BSS) based on time–frequency distributions (TFDs) exploits theunderlying diagonal or off-diagonal structure of TFD matrices to separate the source sig-nals. In this paper, we propose a new signal-independent kernel which is defined in boththe time–lag and the Doppler-lag domain and satisfies most of the desirable propertiesof a TFD. The main objective of this research is to achieve the high resolution and the max-imum cross-term reduction with the preferable diagonal or off-diagonal structure of TFDmatrices in BSS applications. Moreover, a BSS approach is developed which includes firstwhitening mixed signals, then constructing a set of TFD matrices using the proposed TFDand the Hough transform, finally a joint diagonalization of a combined set of TFD matricesto estimate the mixing matrix and the source signals. By use of the techniques proposed inthis paper, the improved performance of BSS of nonstationary signals has been achieved.
� 2011 Elsevier Ltd. All rights reserved.
1. Introduction
The aim of blind source separation (BSS) is to recover the sources form the observations only [1]. Different methods havebeen developed for BSS based on cyclostationary, second-order and/or high-order statistics of the source signals, linear andquadratic time–frequency (t–f) transforms. In many applications such as radar, sonar and acoustic localization, the signals ofinterest are known to be nonstationary in nature. For these signals, BSS based on time–frequency distributions (TFDs) pro-vides an improved performance over other methods, when dealing with signals that are localized in the t–f domain.
Among the existing TFDs-BSS algorithms [2–10], typically, the TFD matrices are constructed from the auto-terms andcross-terms of the observed signals and used for source diagonalization and antidiagonalization, or a combination of bothtechniques, depending on the t–f point selections and the structure of the source TFD matrices. These existing methods,hence, allow the separation of source signals with the identical spectral shape but with the different t–f localizations. How-ever, for multicomponent signals, the inherent bilinear structure of Cohen’s class, for example, the Wigner-Ville Distribution(WVD) and/or its expansions causes the interfering cross-terms which often lead to misinterpretation of the signals.
Several different approaches [11–15] have been proposed to eliminate the cross-terms interference based on time–fre-quency ratio of mixtures (TIFROM). These approaches are inspired by a method that requires single source to occur alonein a tiny t–f zone, i.e. they set very limited constraints on the source sparsity and overlapping. TIFROM and its expansionsrely on identifying the columns of the mixing matrix by first finding the t–f zones where only one source exists and thenindependently estimating the scale coefficients and the time shifts in these single-source zones using ratios of mixtures. TI-
. All rights reserved.
to Editor-in-Chief by Guest Editor Dr. Yi Wan.nd Information Engineering, Southwest University, Chongqing 400715, China. Tel.: +86 23 68117847.
176 J. Guo et al. / Computers and Electrical Engineering 38 (2012) 175–184
FROM can avoid the cross-term interference by use of the short time fourier transform (STFT). The main drawback of TIFROMis that the length of the window, that is, the length of the assumed stationary is directly related to the frequency resolutionand hence has a fixed low resolution, which means that the non-stationary occurring during this interval will be smeared intime and frequency.
The purpose of this paper is to introduce a signal-independent kernel for the design of a new TFD with a high resolutionand a maximum cross-term reduction in the t–f domain. This TFD can be used to improve the BSS performance by accuratelyestimating the instantaneous frequency of multicomponent signals. In this paper, we review the fundamental theory of thecross-term elimination using the ambiguity domain filtering, and then the new kernel for a quadratic TFD is proposed. Thenew kernel is defined in both the time–lag and the Doppler–lag domain, and it satisfies the important properties of a TFD.The commonly used TFDs, namely, the WVD, the STFT, the Zhao-Atlas-marks (ZAM), the Born-Jordan distribution (BJD) andthe Choi-Williams distribution (CWD) are investigated in the comparison of the proposed TFD using the Boashash’s perfor-mance indicator. It shows that the proposed method performs better than the above TFDs in the analysis of multicomponentsignals.
Furthermore, a BSS approach is developed by firstly whitening mixed signals, secondly constructing a set of TFD matricesusing the proposed TFD and the Hough transform and finally a joint diagonalization of a combined set of TFD matrices toestimate the source signals and the mixing matrix. The well-known BSS algorithms, such as the second-order blind identi-fication (SOBI), the fourth order blind identification with transformation matrix (FOBI-E), the fixed-point Independent Com-ponent Analysis (FPICA), the TFD-BSS (using WVD) and the TIFROM (using STFT) are employed in the comparison of theproposed approach. This approach shows a number of attractive features. In contrast to BSS approaches using statisticalinformation of the sources (e.g., the SOBI, the FOBI-E and the FPICA), it allows the separation of Gaussian sources withthe identical spectral shape but with the different t–f localizations. Compared with the BSS approaches based on t–f trans-forms (e.g., the WVD and the STFT), it has the improvements in t–f resolution and energy concentration, and hence has abetter separation performance. Moreover, it has the advantage to spread the noise power while localize the source energyin the t–f domain using Hough transform. This advantage increases the robustness of this algorithm in the presence of addi-tive noise.
This paper is organized as follows. Section 2 presents the data model. The proposed kernel and TFD are described in Sec-tion 3. The BSS algorithm is derived in Section 4. In Section 5, three examples are demonstrated for the evaluation of theproposed technique. Finally, conclusions are drawn in Section 6.
2. Data model
We consider the following linear data model for time t ¼ 1;2; . . . ; T
xðtÞ ¼ AsðtÞ þ nðtÞ ð1Þ
where xðtÞ ¼ ½x1ðtÞ; x2ðtÞ; . . . ; xmðtÞ�T is a vector of observations, sðtÞ ¼ ½s1ðtÞ; s2ðtÞ; . . . ; snðtÞ�T is a vector containing the non-stationary sources. It is assumed the sources are zero-mean and mutually uncorrelated. A is a full rank mixing matrix witha dimension of m� n where m P n, nðtÞ is an additive noise vector. Its statistics is defined by
Efnðt þ sÞnðtÞg ¼ dðsÞr2Im ð2Þ
where Im is the m� n identity matrix, r2 is the variance of the noise and dðsÞ is the Dirac function. The covariance matrices ofvectors sðtÞ and xðtÞ are:
where qiðt; sÞ ¼ Efsiðt þ sÞs�i ðtÞg denotes the autocovariance of siðtÞ, and the superscript H represents the conjugatetranspose.
3. Proposed TFD
There are many TFDs. Each has its own merits and drawbacks. The choice of a particular TFD depends on the represen-tation properties that are desirable for that special application. In general, TFD-BSS and/or TIFROM require that the TFD usedin the BSS should be cross-terms free and should have a high-resolution in the t–f domain. Since they simplify the selectionof t–f points that enter the joint diagonalization procedure, this is crucial to the performance of BSS. Unfortunately, these twoconditions seem to be conflicting. For example, the WVD is known to have a high resolution in both time and frequency,however, it suffers from the presence of cross-terms for the multicomponent signals. The STFT is cross-terms free, but ithas a fixed low t–f resolution. In order to improve the BSS performance, we need a new distribution which gets rid of thecross-terms and preserves the high t–f resolution in order to achieve the preferable diagonal or off-diagonal structure ofTFD matrices. In this section, the fundamental concepts of the cross-term elimination using the ambiguity function are re-viewed and then the new t–f kernel is proposed.
J. Guo et al. / Computers and Electrical Engineering 38 (2012) 175–184 177
3.1. Proposed kernel
The Cohen’s class TFDs of a signal xðtÞ can be formulated by
qxxðt; f Þ ¼Z þ1
�1
Z 1
�1gðv ; sÞAxxðv ; sÞej2pðvt�fsÞdvds ð5Þ
where gðv ; sÞ is a signal-independent kernel function, associated with a particular TFD. The variables t and f denote time andfrequency, respectively. s and v represent the variables of time lag and frequency lag, respectively, in the ambiguity functiondomain. If Axxðv ; sÞ is the ambiguity function of xðtÞ, it is given by
Axxðv ; sÞ ¼Z þ1
�1x t þ s
2
� �x� t � s
2
� �e�j2pvtdt: ð6Þ
Eq. (5) shows that qxxðv ; sÞ may be obtained by firstly multiplying the kernel gðv ; sÞ by the ambiguity function Axxðv; sÞand then carrying out a two-dimensional Fourier transform. Therefore, the key to understand the t–f relationships is throughthe ambiguity domain filtering. Generally, the auto-terms are known to be mapped around the origin in the ambiguity do-main whereas the cross-terms are mapped away from the origin at a distance function of the separation. The distance is di-rectly proportional to the time and frequency distance of the signal components. Therefore, if the kernel function gðv ; sÞ inthe ambiguity domain is properly chosen as a two-dimensional function centered around the origin with the sharp cut-offedges, it can reduce the cross-terms in analysis of multicomponent signals. For instance, the WVD, with an all-pass kernel ofgðv ; sÞ ¼ 1, presents the best t–f concentration and the undesirable high-amplitude cross-terms in the analysis of multicom-ponent signals. Another example is the STFT. It is traditionally used for the t–f analysis of speech signals. The distribution isdefined by selecting the kernel as the ambiguity function of an arbitrary window function.
The kernel function gðv ; sÞ is defined as
gðv ; sÞ ¼Z þ1
�1Gðt; sÞe�j2pvtdt: ð7Þ
Therefore, the kernel function gðv ; sÞ is the Fourier transform of Gðt; sÞwith respect to time t. Since Gðt; sÞ is defined in thetime–lag domain, we begin to select an exponential kernel e�jsj in the lag domain and then multiply it by a hyperbolic timefunction sech2ðtÞ with a perfect frequency resolution. This extends the kernel to a two-dimensional function. Next, assign itto a power of a. Consequently, the proposed separable kernel, defined in the time–lag domain, can be expressed as
Gðt; sÞ ¼ ðe�jsjsech2ðtÞÞa ð8Þ
where a is a real parameter that controls the sharpness of the cut-off frequency of the filter and hence the trade-off betweenthe cross-terms elimination and the t–f resolution. The value of a ranges between zero and unity, that is, (0 < a < 1Þ. It isworthwhile to note that the filtering of the cross-terms in the t–f domain unfortunately results in a low t–f resolution. Thismeans that there is a balance between the cross-terms suppression and the high t–f resolution.
3.2. Proposed TFD and its properties
Substituting Eq. (8) into (7), the Doppler-lag kernel gðv; sÞ is given in the following form
gðv ; sÞ ¼Z þ1
�1ðe�jsjsech2ðtÞÞae�j2pvtdt ¼ 2e�ajsj
Z þ1
0sech2aðtÞ � cosð2pvtÞdt: ð9Þ
By use of the formula 3.985.1 in [16], that is,
Z þ1
0sechmðbtÞ � cosðatÞdt ¼ 2m�2
bCðmÞCm2þ ja
2b
� �C
m2� ja
2b
� �for ½Re b > 0;Re m > 0;a > 0�; ð10Þ
the kernel gðv; sÞ is given by
gðv ; sÞ ¼ e�ajsj 22a�1
Cð2aÞCðaþ jpvÞCða� jpvÞ: ð11Þ
Consequently, the proposed TFD, called G-distribution, is expressed by
GTxxðt; f Þ ¼Z þ1
�1
Z þ1
�1e�jsjsech2ðvÞ� �a
xðt þ v þ s2Þx� t þ v � s
2
� �e�j2pfsdvds ð12Þ
where superscript ⁄ denotes the complex conjugate. Ideally, there is a set of desirable properties that a TFD should satisfy[17,18]. In particular, the proposed TFD satisfies the following properties:
(1) Realness: It satisfies the realness properties since gðv ; sÞ ¼ g�ð�v;�sÞ.(2) Time shift invariance: It satisfies the time shift invariance properties since gðv ; sÞ dose not depend on time t.
178 J. Guo et al. / Computers and Electrical Engineering 38 (2012) 175–184
(3) Frequency shift invariance: It satisfies the frequency shift invariance properties since gðv ; sÞ is not a function of fre-quency f.
(4) Reduced interference and resolution: It satisfies the properties as it has a high resolution with a cross-term reduc-tion as shown in Section 5 for the case of two linear frequency modulation (LFM) signals.
The above properties are the most important ones needed to achieve a high t–f resolution and a high separation perfor-mance for the BSS applications. The evaluation criteria and the performance simulations are discussed in Sections 3.3 and 5.
3.3. Choice of a
The choice of a in Eq. (12) is application dependent. Its optimal value can be determined by minimizing the Boashash’sperformance indicator P. The indicator P is defined as [19]
P ¼ jASjjAXjjAMj2D
P 0 ð13Þ
where AM, AS and AX are the average amplitudes of the mainlobes, the sidelobes and the cross-terms, respectively, of anytwo consecutive components of the multicomponent signals, as illustrated in Fig. 1.
The relative position separation D between the components’ mainlobes about their respective instantaneous frequency,f 1 and f 2, is expressed as:
D ¼ 1� V1þ V22ðf 2� f 1Þ ð14Þ
where V1 and V2 are the 3.0 dB mainlobe bandwidths of the first and the second component, respectively.The indicator P provides a metric of the resolution capability, which takes into account the key attributes of TFDs, such as
component mainlobes, sidelobes and cross-terms. Eq. (13) indicates that the best performance of TFD is achieved by mini-mizing P. Therefore, we choose the indicator P to optimize the value of a for the proposed TFD. It includes three procedures:Firstly, we choose an initial value of a between zero and one, usually the smallest value, and calculate the TFD for the givensignal; Secondly, we take the consecutive slices of this TFD, find the indicator P for each of the slices, and average all suchobtained indicators P to obtain the average performance indicator Paverage; Thirdly, repeating this procedure over a range of(0,1), the optimal value of a is estimated by identifying the one which results in the smallest Paverage. In this paper, we havefound that when a is equal to 0.012 the best trade-off is achieved as shown in Section 5.
0 0.1 0.2 0.3 0.4 0.5-0.5
0
0.3
0.6
0.9
1.2
Normalized frequency
Nor
mal
ized
am
plitu
de
Slice of a TFD of a two-component signal
AX
AM2AM1
AS2AS1
V2
f2f1
V1
Fig. 1. Slice of a TFD of a two-component signal.
J. Guo et al. / Computers and Electrical Engineering 38 (2012) 175–184 179
4. Blind separation approach
In this section, we present a BSS approach involving three steps: Firstly whitening mixed signals; secondly constructing aset of TFD matrices using the proposed TFD and the Hough transform; finally a joint diagonalization of a combined set of TFDmatrices to estimate the mixing matrix and the source signals.
Step 1: Whitening the mixed signals: This objective of this step is to transform the mixing matrix A into a unitary matrix U.This is achieved by searching for a n�m whitening matrix W such that U ¼WA and WAAHWH ¼ In. The matrix W can beestimated by [3]
W ¼ ðk1 � r2Þ�12h1; . . . ; ðkn � r2Þ�
12hn
h ið15Þ
where [k1; . . . ; kn] are the n largest eigenvalues of Rxxðt;0Þ sorted in decreasing order and [h1; . . . ;hn] are the correspondingeigenvectors. r2 can be estimated by the mean value of the m� n smallest eigenvalues of Rxxðt;0Þ as knþ1 ¼ � � � ¼ km ¼ r2
where knþ1; . . . ; km are the m� n smallest eigenvalues.Step 2: Constructing TFD matrices of whitening signals: The aim of this step is to obtain a set of different localization prop-
erty of whitening signals. This is achieved by calculating the TFD and Hough transform of whitening signals to localize thesignals energy in the noise background, and hence it is given by
GHTxxðf ; gÞ ¼Z þ1
�1
Z þ1
�1GTxxðt;vÞdðv � f � gtÞdtdv ¼
Z þ1
�1GTxxðt; f þ gtÞdt: ð16Þ
Eq. (16) can be interpreted as a line integral of the G-distribution in the parameter space. The parameters corresponding toxðtÞ are [20]
g ¼ � cotðhÞ; f ¼ q= sinðhÞ and q ¼ x cosðhÞ þ y sinðhÞ ð17Þ
where q is the distance from the origin of the image to the line, and h is the angle between the normal to the line and the taxis. The points in the ðt; f Þ space are mapped to the ðq; hÞ space by Eq. (17), each point corresponds to a sine curve. The pur-pose of the mapping is to find the imperfect instances of objects within a certain class of shapes by a voting procedure. Thisvoting procedure is executed in a parameter space, from which the object candidates are obtained as the local maximum inan accumulator space that is explicitly constructed by the algorithm to compute the Hough transform [21].
Step 3: Joint-diagonalization a set of GHT matrices: The objective of this step is to jointly diagonalize a set of TFD matricesand hence to retrieve the unitary matrix U. Given estimates W, we get
GHTxxðf ; gÞ ¼ A� GHTssðf ; gÞ � AH þ r2Im ð18Þ
Considering a low noise environment, the noise in Eq. (18) is neglected, that is,
GHTxxðf ; gÞ � A� GHTssðf ; gÞ � AH ð19Þ
By pre and post multiplying GHTxxðf ; gÞ by W, we have
Since GHTzzðf ; gÞ is known, Eq. (20) shows that U may be obtained as a joint diagonalizing matrix of a set of whitened TFDmatrices [22]. The joint diagonalizaiton allows the information contained in a set of t–f matrices to be integrated in a singleunitary matrix U, and it is executed by a generalization of the Jacobi technique.
The obtained W and U from the above steps are then used to estimate the source signals sðtÞ and the mixing matrix A withthe relations s ¼ UHWxðtÞ and A ¼ W#U, where the superscript # denotes the Moore–Penrose pseudoinverse. A block dia-gram of the proposed three-stage approach is shown in Fig. 2.
Fig. 2. Block diagram of the proposed GHT-BSS approach.
180 J. Guo et al. / Computers and Electrical Engineering 38 (2012) 175–184
5. Numerical experiments
In this section, three experiments were conducted using the simulated and the real data to investigate the t–f resolutionof the proposed algorithm and its separation performance.
Experiments 1: Multi-component LFM signals: In the first example, we used the Boashash’s performance indicator P to com-pare the performance of the WVD, the STFT, the ZAM, the BJD, the CWD and the proposed TFD (a ¼ 0:012Þ. The multicom-ponent signal is expressed by
For each TFD, we take a slice in the middle of the time interval with a total length of N ¼ 128 and a sampling interval ofT ¼ 1 and measure the parameters including AM, AS, AX and V. These parameters are then used to calculate the relative posi-tion separation D, defined by Eq. (14), and the indicator P, defined by Eq. (13). The distributions and their respective mea-surements are recorded in Table 1. In particular, the slices of the WVD, the STFT and the proposed TFD in the middle of thetime interval are displayed in Fig. 3.
Eq. (13) suggests that a TFD, at a given time instant, having the smallest positive value of the measurement P is the TFDwith the best resolution performance at that time instant for the signal under analysis. Table 1 shows that the proposed TFDhas the smallest value of PðP ¼ 0:000109Þ and hence it is regarded as the best. Similar results are obtained with other typesof signals. Fig. 3 clearly indicates the superiority of the proposed TFD over WVD in terms of the cross-term suppression andthe superiority of the proposed TFD over STFT in terms of the t–f resolution. The superiority of the proposed TFD benefitsfrom the sharp cut-off edges of its kernel in the ambiguity domain, and hence it separates each component of the signal moreprecisely.
Experiments 2: Multi-Gaussian sources: In the second example, we compared the BSS performance of the TFD-BSS, the TFI-ROM, the SOBI, the FPICA, the FOBI-E and proposed TFD ða ¼ 0:012Þ using Gaussian signals. The performances were evalu-ated using the signal to interference ratio (SIR) and the performance index (PI) as they are commonly employed to assess theBSS algorithms. As a basic index for the evaluation, SIR is defined as
SIRi ¼ �10log10jjsiðtÞ � siðtÞjj22jjsiðtÞjj22
( )ð22Þ
where siðtÞ and siðtÞ are the estimated signal and the source signal, respectively. SIR is a parameter measuring the similarlevel of the separated signal and the source signal. Higher value of SIR is desirable. In general, when SIR > 26 dB, an idealBSS result is achieved. Another parameter PI, the most widely used measurement to assess the accuracy of the estimatedmixing matrix, is defined by
PI ¼ 1NðN � 1Þ
XN
i¼1
XN
k¼1
jgikjmax
jjgijj� 1
0@
1Aþ XN
k¼1
jgkijmax
jjgjij� 1
0@
1A
8<:
9=; ð23Þ
where gij is the ði; jÞth element of global matrix G ¼ WA. maxjjgijj is the maximum value in the ith row and max
jjgjij is the
maximum value in the ith column. The PI is a parameter measuring the global matrix G suitable for the degeneracy condi-tions. Low value of the PI is desirable. Ideally it should be zero.
Three fourth-order colored sources were taken from the web page of Prof. Andrzej Cichocki at the Laboratory for Ad-vanced Brain Signal Processing, RIKEN Brain Science Institute, Tokyo, Japan. These signals are the rather ‘‘difficult’’ bench-marks. The mixing matrix was set to: A = [0.1509 0.8600 0.4966; 0.6979 0.8537 0.8998; 0.3784 0.5936 0.8216]. The plotsof the source signals are shown in Fig. 4 and the mixed signals are displayed in Fig. 5. Fig. 6 shows the estimated signals.It is clear that the new algorithm works well in this case. Table 2 shows the compared results evaluated by 30 Monte-Carlosimulations.
In this case, the TIFROM (using the STFT) has failed to separate these sources as the single-source analysis zones could notbe detected. The SOBI, the FPICA, and FOBI-E using statistical information available on the source signals have also failed.This is due to the fact that the original and mixed distributions are identical and thus there is no way to infer the mixingmatrix from the observed signals [23]. Table 2 indicates that the proposed method achieves the higher SIR (SIR = 26.2374)
asurements and the performance indicator P of TFDs for two LFMs.
Fig. 3. Slices in the middle of the time interval of TFDs for two LFMs.
200 400 600 800 1000 1200 1400 1600 1800 2000-202
Time
Source 1
200 400 600 800 1000 1200 1400 1600 1800 2000-2
0
2
Time
Source 2
200 400 600 800 1000 1200 1400 1600 1800 2000-2
0
2
Time
Source 3
Fig. 4. The plots of fourth order colored sources.
J. Guo et al. / Computers and Electrical Engineering 38 (2012) 175–184 181
and the lower PI (PI = 0.07582) comparing with TFD-BSS (WVD with SIR = 19.2098 and PI = 0.1432). The improvement in theperformance mainly results from the high t–f resolution of the proposed TFD, and hence the TFD can simplify the selection oft–f points that enter the joint diagonalization procedure. The choice of the t–f points has a direct impact on the separationperformance. Therefore, the proposed method can separate the Gaussian sources with the identical spectral shapes but withthe different t–f localizations.
Example 3: Two speech signals highly correlated in the presence of additive Gaussian noise: In the third example, we assessedthe robustness of the TFD-BSS, the TFIROM, the SOBI, the FPICA, the FOBI-E and the proposed TFD ða ¼ 0:012Þwith respect tonoise using SIR index defined by Eq. (22). The microphones were deployed in a room with a size of 3:2� 3:4� 2:8 m. Twohighly correlated speech signals were produced by two speakers to read the same sentence. They are selected as the originalsources, which are sampled at a frequency of 8 kHz. These sources are mixed by the following matrix A ¼ ½10:9; 0:81� and
Fig. 7. SIR curve of the estimated sources versus SNR.
J. Guo et al. / Computers and Electrical Engineering 38 (2012) 175–184 183
then are corrupted by the additive Gaussian noise with SNR ranges from 30 dB to 100 dB. Fig. 7 shows the average SIR curveof these algorithms under different additive Gaussian noise environments.
Fig. 7 indicates that the performance of all separation algorithms degrades with the decrease of SNR significantly becausethe observed mixtures are corrupted by additive Gaussian noise and hence the mixed signals do not give us a direct access tothe source signals, as in noiseless BSS. That is the reason why BSS problem under noise is known as a hard problem. Fig. 7shows that our algorithm has a better estimating performance than other methods under almost all tested environmentalconditions. The increase of this robustness of our algorithm with respect to noise mainly results from the high t–f resolutionof the proposed TFD and from the effect of spreading the noise power while localizing the source energy by Hough transform.Hence we can select a set of ‘‘clean’’ spatial t–f points that represent the true signal power to estimate the source signal sðtÞand the mixing matrix A.
6. Conclusions
In this paper, we have investigated several important issues in TFDs-based BSS problems. First, a novel member of thequadratic TFD has been introduced. The kernel of this TFD has been defined in both the time–lag and the Doppler–lag domainand it has many desirable properties. It has been shown, using the sum of two LFM signals, that the proposed TFD hasachieved a high resolution performance in the t–f plane. The quantitative comparative results have also confirmed the supe-riority of this TFD in terms of the trade-off between the cross-term suppression and the high energy concentration in the t–fdomain. The t–f resolution capability of this TFD has a direct impact on the BSS performance.
Second, a BSS approach using the new TFD has been proposed. The approach has used a new ‘clean’ TFD to obtain the t–fmatrices of mixed signals, then localized the source energy by Hough transform and obtained the estimated signals based onthe diagonalization of a combined set of TFD. It has been shown, using the synthetic and real multicomponent signals, thatthe proposed algorithm has achieved the highest SIR and the lowest PI performance comparing with other methods. Thisapproach has also increased the BSS robustness by spreading the noise power and localizing the source energy in the t–f do-main. The improvement of performance mainly results from the high t–f resolution of the proposed TFD, and hence themethod can separate nonstationary sources with the identical spectral shapes but with the different t–f localizations.
A drawback of the proposed algorithm is that the computational complexity grows exponentially as the number of obser-vations in the t–f plane increases. One approach to overcome this defect is to reduce the maximum number of active sourcesbased on the sparsity of signals.
Acknowledgments
We would like to express our sincere thanks for the valuable suggestions and comments made by the reviewers and thesupport of the National Natural Science Foundation of China (Grant No. 60971016). We would also like to thank Prof. Andrzej
184 J. Guo et al. / Computers and Electrical Engineering 38 (2012) 175–184
Cichocki from the Lab for Advanced Brain Signal Processing, RIKEN Brain Science Institute, Tokyo, Japan, to provide the dataused in the second example.
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Jing Guo received the B.S. degree in Electrical Engineering from the Wuhan University in 1998, the M.S. degree in Communication Engineering form theWuhan University of Technology in 2004. He is currently a Ph.D. candidate in Communication Engineering at the Chongqing University. His researchinterests are in the areas of time–frequency analysis and blind source separation.
Xiaoping Zeng received the Ph.D. degree in Electrical Engineering at the Chongqing University in 1996. He is currently a professor and dean of College ofCommunication Engineering at the Chongqing University. His research interests span signal processing, neural network and their applications to real-worldproblems.
Zhishun She received the Ph.D. degree in Electrical and Electronic Engineering from the University of Adelaide, Australia in 2000. Currently, he is a reader atEngineering Department, Glyndwr University, University of Wales, UK. His research interests include signal and image processing with applications toremote sensing, radar and medical diagnosis.
