MSC 2010: 42C40, 94A12On the blind source separation problem, there is a method to use the quotient function of complex valued time-frequency informations of two ob-served signals. By studying the quotient function, we can estimate the number of sources under some assumptions. In our previous papers, we gave a mathematical formulation which is available for the sources with-out time delay. However, in general, we can not ignore the time delay. In this paper, we will reformulate our basic theorems related to the method of estimating the number of sources to be available for more general cases

Fujita, K.

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

ON THE QUOTIENT FUNCTION EMPLOYED INTHE BLIND SOURCE SEPARATION PROBLEMK. FujitaAbstractOn the blind source separation problem, there is a method to use thequotient function of complex valued time-frequency informations of two ob-served signals. By studying the quotient function, we can estimate thenumber of sources under some assumptions. In our previous papers, wegave a mathematical formulation which is available for the sources with-out time delay. However, in general, we can not ignore the time delay. Inthis paper, we will reformulate our basic theorems related to the method ofestimating the number of sources to be available for more general cases.MSC 2010: 42C40, 94A12Key Words and Phrases: quotient function, cumulative distributionfunction, blind source separation1. IntroductionTo treat blind source separation problem, in many cases, either statis-tical independence or statistical orthogonality of the sources has been as-sumed. Around 2000, papers as [4], [9], etc., considered the blind source sep-aration problem under some independence of the windowed Fourier trans-forms of the sources in the time-frequency domain. To solve the blindsource separation problem, as the first step, they try to detect the numberc© 2010, FCAA – Diogenes Co. (Bulgaria). All rights reserved.496 K. Fujitaof sources. The fundamental idea to detect the number of sources employedin [4], [9], [10], etc., is to consider the ratio of the windowed Fourier trans-forms of the two observed signals. In [1], [7] and [8], we gave a mathematicalformulation for this estimation problem of the number of sources withouttime delay. In [5] and [6], we gave some remarks on our method for appli-cations. Later, [2] and [3] treated the problem with time delay and gave analgorithm of the numerical experiment.In this paper, we will reformulate our results in [1] so that we can applyto more general cases.2. A method of blind source separationLet n ≥ 2 be an unknown integer and xk(t) be an observed signalof unknown sources {sj}nj=1. We assume that observed signals {xk} arerepresented asxk(t) =n∑j=1ajksj(t− cjk), (1)where ajk are unknown real numbers and cjk > 0.A method of blind source separation is as follows:1. Transfer x1 and x2 to complex valued continuous functions X1(t, ω)and X2(t, ω) in a time-frequency domain under suitable transforma-tion.2. Consider the quotient function Q(t, ω) = X1(t, ω)/X2(t, ω). By study-ing Q(t, ω), estimate the number of sources (say n) under some as-sumptions.3. Take another (n− 2) observed signals and determine ajk, cjk, and theoriginal sources sj , j, k = 1, · · · , n.3. Mathematical formulationLet d, n ∈ N, X = Rd, d ≥ 2, or X = Cd, d ≥ 1, and Sj(z), j =1, 2, · · · , n, are linearly independent as complex valued continuous functionson X. For aj , bj ∈ R \ {0} and cjl = (cjl,1, cjl,2, · · · , cjl,d) ∈ X, l = 1, 2, setX1(z; c1) =n∑j=1ajSj(z − cj1), X2(z; c2) =n∑j=1bjSj(z − cj2),ON THE QUOTIENT FUNCTION EMPLOYED IN . . . 497X1(z; 0) =n∑j=1ajSj(z), X2(z; 0) =n∑j=1bjSj(z).3.1. Complex valued quotient functionsConsider the quotient functionQ(z; c)=X1(z; c1)X2(z; c2)=a1S1(z − cj1) + ...+ anSn(z − cj1)b1S1(z − cj2) + ...+ bnSn(z − cj2) , Q(z; 0)=X1(z; 0)X2(z; 0).For η > 0, we define the following function Qη(z; c) which depends on thevalues of =Q, the imaginary part of Q:Qη(z; c) ={Q(z; c) (|=Q(z; c)| < η),0 (|=Q(z; c)| ≥ η).In the case of cj1 = cj2 = 0, j = 1, · · · , n, Q(z; c) takes real valueqj ≡ aj/bj onEj = {z ∈ X;Sj(z) 6= 0, Sk(z) = 0, (k 6= j)}.Let us introduce some notations. PutDj = {z ∈ X; Sj(z) 6= 0},Dcj = Dj⋃{z ∈ X; z − cj1 ∈ Dj}⋃{z ∈ X; z − cj2 ∈ Dj},D =n⋃j=1Dcj , E =n⋃j=1Ej .For M > 0, we setB(M) = {z = (z1, · · · , zd); |zl| < M, l = 1, · · · , d}, D(M) = D ∩B(M),Ej(M) = Ej ∩B(M), E(M) = ∪nj=1Ej(M), Ec(M) = D(M) \ E(M).If cj1 = cj2 = 0, j = 1, · · · , n, D = E and Ej 6= ∅, we can easilydetect the number of {Sj} by counting the number of elements of the imageQη(D) = Q(D) = Q(E).498 K. Fujita3.2. Cumulative distribution function on RWe denote the Lebesgue measure of a measurable set A ∈ X byµ(A) =∫Adz,where dz is the Lebesgue measure on A and use the following notation:νc(A) = µ({z ∈ A; =Q(z; c) = 0, Q(z; c) 6= 0}).We consider a function (Gη(M ; c))(x) that describes the distributionof values of <Qη(z; c), the real part of Qη(z; c). For (Gη(M ; c))(x) to bewell-defined, we assume the following condition throughout this paper.νc(D(M)) > 0.For η > 0,M > 0, and x ∈ R, (Gη(M ; c))(x) is defined as follows:(Gη(M ; c))(x) =µ({z ∈ D(M); <Qη(z; c) < x, Qη(z; c) 6= 0})µ({z ∈ D(M); Qη(z; c) 6= 0}) . (2)By the definition (Gη(M ; c))(x) is a monotone increasing function. Furtherwe define(G0(M ; c))(x) =µ({z ∈ D(M); <Q(z; c) < x, =Q(z; c) = 0, Q(z; c) 6= 0})νc(D(M)).Similar to Theorem 1 in [1], we can prove|(Gη(M ; c))(x)− (G0(M ; c)(x)| ≤ µ({z ∈ D(M); 0 < |=Qη(z; c)| < η})µ({z ∈ D(M); Qη(z; c) 6= 0})≡ βη(M ; c) (3)and limη→0(Gη(M ; c))(x) = (G0(M ; c))(x).3.3. In the case of D = EFirst we consider the case that cj1 = cj2 = 0, Ej(M) 6= ∅, j = 1, · · · , nand D = E. In this case, (Gη(M ; 0))(x) is the step function (see Fig. 1):H0(x) ≡ (Gη(M ; 0))(x) =n∑j=1ν0(Ej(M))ν0(E(M))Y (x− qj) , Y (x) ={1 (x > 0),0 (x ≤ 0).ON THE QUOTIENT FUNCTION EMPLOYED IN . . . 499Thus we can detect the number of {Sj} by counting the number of thesteps which appear on the graph of (Gη(M ; c))(x). In numerical analysis,the graph ofG′η(M ; c) = d(Gη(M ; c))(x)/dx, where the derivative is taken inthe sense of distribution, is often used. Since the derivative of the Heavisidefunction Y (x) is the delta function, we may count the number of the peakswhich appear on the graph of G′η(M ; c). In the following we assume q1 <q2 < · · · < qn.Fig. 1(Gη(M ; c))(x)6- xq1ppq2pppppq3ppppppp=⇒(G′η(M ; c))(x)q16q26q36Next we consider the case that∑j(c2j1 + c2j2) 6= 0; (that is, at least onecjk does not vanish), Ej(M) 6= ∅, j = 1, · · · , n, and D = E. Since we assumethat Sj(z) are continuous, Sj(z) ≈ Sj(z− cjk) for sufficiently small cjk. PutSj(z − cjk)− Sj(z) = ξjkeiθjk , ξjk > 0, θjk ∈ [−pi, pi], ξ1 =∑nk=1 akξk1eiθk1and ξ2 =∑nk=1 bkξk2eiθk2 . Then on the set Ej(M),<Qη(z; c)− qj = <ξ1 + ajSj(z)ξ2 + bjSj(z)− ajbj= < bjξ1 − ajξ2bj(ξ2 + bjSj)= < ξ1 − qjξ2(ξ2 + bjSj).DefineHc(x) =n∑j=1νc(Ej(M))νc(E(M))Y (x− qj),(g(M ; c))(x) =µ({z ∈ E(M);<Q(z; c) < x,=Q(z; c) = 0, Q(z; c) 6= 0})νc(E(M)).Then similar to Theorem 2 in [1], by using Lemma 2 in [1] we have|G0(M ; c)(x)− g(M ; c)(x)| ≤ ν(Ec(M))νc(D(M))≡ β0(M ; c). (4)Since we assume that D = E, β0(M ; c) = 0. Thus by (3) and (4) we have|(Gη(M ; c))(x)− (g(M ; c))(x)|= |(Gη(M ; c))(x)− (G0(M ; c))(x) + (G0(M ; c))(x)− (g(M ; c))(x)|≤ βη(M ; c) + β0(M ; c) = βη(M ; c).500 K. FujitaAt last, we consider the difference between (g(M ; c))(x) and Hc(x). Putmaxz∈Ej|Qη(z; c)− qj | = ej , q+j = qj + ej , q−j = qj − ej .Assume γ ≡ maxj ej < minj 6=k |qj − qk|/2. Then similar to Proposition 2 in[1] we can prove that(g(M ; c))(x) = Hc(x), x ∈ R \⋃j[q−j , q+j ],Hc(x− γ) ≤ (g(M ; c))(x) ≤ Hc(x+ γ), x ∈⋃j[q−j , q+j ].Thus puttingH±(x) = n∑j=1νc(Ej(M))νc(E(M))± βη(M ; c)Y (x− q∓j ) ,we have the following Theorem:Theorem 1. Let D = E, νc(E(M)) > 0 and Ej(M) 6= ∅, j = 1, · · · , n.Assume γ < minj 6=k |qj − qk|/2. Then the graph of the monotone increasingfunction (Gη(M ; c))(x) is contained in the closed domain E = {(x, y) ∈R2; H−(x) ≤ y ≤ H+(x), 0 ≤ y ≤ 1}. (See the following stair like domain(Fig2˙, left)).A graph of (Gη(M ; c))(x) will be as in the right Fig. 2 and is in the stairlike domain E on the left Fig. 2.- xq1q+1q−1p ppppq2q+2q−2ppppppppppppppppq3q+3q−3pppppppppppppppppppppppppppp6(Gη(M ; c))(x)- xq1pppq2pppppppq3ppppppppppFig. 2ON THE QUOTIENT FUNCTION EMPLOYED IN . . . 501Remark. If |minj 6=k |qj − qk| − 2γ| is so small, then it will be hard tosee the steps on the graph of (Gη(M ; c))(x).3.4. In the case of D 6= EWhen D 6= E, then β0(M ; c) 6= 0 and ej will be bigger. Thus the stairlike domain in left Fig. 2 becomes larger (see Fig. 3). Putρη(M ; c) = βη(M ; c) + β0(M ; c),H˜±(x) = n∑j=1νc(Ej(M))νc(E(M))± ρη(M ; c)Y (x− q∓j ) .Then Theorem 1 will be restated as follows:Theorem 2. Let νc(E(M)) > 0 and Ej(M) 6= ∅, j = 1, · · · , n. Assumeγ < minj 6=k |qj−qk|/2. Then the graph of the monotone increasing function(Gη(M ; c))(x) is contained in the closed domain E = {(x, y) ∈ R2; H˜−(x) ≤y ≤ H˜+(x), 0 ≤ y ≤ 1}.Fig. 3- xq1pppppq2ppppppppq3ppppppppppD = E =⇒ D 6= E- xq1pppppq2ppppppppq3pppppppppp4. A generalizationThe assumption νc(E(M)) > 0 in Theorem 1 and Theorem 2 is restric-tive, in our previous papers we introduced the following space:Ej(δ;M) = {z ∈ D(M); |bkSk(z)| ≤ δ|bjSj(z)| (k 6= j), Sj(z) 6= 0} .502 K. FujitaRoughly speaking, Ej(δ;M) describes the set of points where Sj(z) is domi-nant over the other Sk(z)’s (k 6= j). In this paper we introduce the followingspaces:Ecj1(δ) = {z ∈ D(M); |bkSk(z − ck1)| ≤ δ|bjSj(z)| (k 6= j),≤ δ1|bjSj(z − cj1)|, Sj(z) 6= 0},Ecj2(δ) = {z ∈ D(M); |bkSk(z − ck2)| ≤ δ|bjSj(z)| (k 6= j),≤ δ2|bjSj(z − cj2)|, Sj(z) 6= 0},Ecj (δ;M) = Ecj1(δ)⋂Ecj2(δ),E(δ, c;M) = ∪jEcj (δ;M), Ec(δ, c;M) = D(M) \ E(δ, c;M).If δ2 < 1/(n− 1), then on the set Ecj (δ;M), we have|Qη(z)− qj | =∣∣∣∣∑nk=1 akSk(z − ck1)∑nk=1 bkSk(z − ck2)− qj∣∣∣∣≤ |∑nk=1 bk(qkSk(z − ck1)− qjSk(z − ck2))||bjSj(z − cj2)| −∑k 6=j |bkSk(z − ck2)|≤ |∑nk=1 bk(qkSk(z − ck1)− qjSk(z − ck2))|δ/δ2|bjSj(z)| − (n− 1)δ|bjSj(z)|=|∑nk=1 bk ((qk − qj)Sk(z − ck1) + qj(Sk(z − ck1)− Sk(z − ck2))) |δ/δ2|bjSj(z)| − (n− 1)δ|bjSj(z)|≤ (n− 1)δ2∆1− (n− 1)δ2 +∆′ δ2δ|∑nk=0 bk(Sk(z − ck1)− Sk(z − ck2))|(1− (n− 1)δ2)|bjSj(z)| ,where we put ∆ = maxj,k |qj − qk| and ∆′ = maxj |qj |. Further we putCj = maxz∈Ecj (δ;M)|∑nk=0 bk(Sk(z − ck1)− Sk(z − ck2))||bjSj(z)| ,γ(δ) =(n− 1)δ2∆1− (n− 1)δ2 , γ(c) =δ2δ∆′maxj Cj1− (n− 1)δ2 , γ(δ, c) = γ(δ) + γ(c).If Sj(z − cj1) = Sj(z − cj2), j = 1, · · · , n, then γ(c) = 0. Further if cj1 =cj2 = 0, j = 1, · · · , n, then we can take δ = δ1 = δ2 besides γ(c) = 0. Define(gδ(M ; c))(x)=µ({z∈E(δ, c;M);<Q(z; c)<x,=Q(z; c)=0, Q(z; c) 6=0})νc(E(δ, c;M)),ON THE QUOTIENT FUNCTION EMPLOYED IN . . . 503Hc,δ(x) =n∑j=1νc(Ecj (δ;M))νc(E(δ, c;M))Y (x− qj)) .Then similarly to Theorems 1 and 2 in [1], we can prove|Gη(M ; c)(x)− (gδ(M ; c))(x)|= |Gη(M ; c)(x)− (G0(M ; c))(x) + (G0(M ; c))(x)− (gδ(M ; c))(x)|≤ βη(M ; c) +νc(Eccj (δ;M))νc(D(M))≡ ρη(δ, c;M),where βη(M ; c) is defined by (3). Further similar to Proposition 2 in [1],under the assumption of γ(δ, c) < minj 6=k |qj − qk|/2, we can prove that(gδ(M ; c))(x) = Hc,δ(x), x ∈ R \ U [q; γ(δ, c)],Hc,δ(x− γ(δ, c)) ≤ (gδ(M ; c))(x) ≤ Hc,δ(x+ γ(δ, c)), x ∈ U [q; γ(δ, c)],where we put U [q; γ(δ, c)] =⋃j [qj−γ(δ, c), qj +γ(δ, c)]. Define the two stepfunctions by(H±c,η(δ;M))(x) = n∑j=1νc(Ecj (δ;M))νc(E(δ, c;M))± ρη(δ, c;M)Y (x− (qj ∓ γ(δ, c))) .From the above-mentioned consideration, the following theorem is ageneralization of Theorem 3 in [1]:Theorem 3. Let η > 0, 1/(n − 1) > δ2 > 0 and let (Gη(M ; c))(x)be the function defined by (2). We assume that δ and δ2 are chosen sosmall that γ(δ, c) < minj 6=k |qj − qk|/2 is satisfied. Further, we assumeνc(E(δ, c;M)) > 0. Then the graph of the monotone increasing function(Gη(M ; c))(x) is contained in the closed domain:{(x, y) ∈ R2; (H−c,η(δ, c;M))(x) ≤ y ≤ (H+c,η(δ, c;M))(x), 0 ≤ y ≤ 1}.- xq1pppppq2ppppppppq3ﬀ- γ(δ, c)?6ρη(δ, c;M)pppppppppp(H+c,η(δ;M))(x) ↓ ← (H−c,η(δ;M))(x)(Gη(M ; c))(x)↖Fig. 4504 K. FujitaWhen X = R2 and cj1 = cj2 = 0, j = 1, · · · , n, then this theorem is justTheorem 3 in [1].3.4. Remarks on Theorem 31. γ(δ) is an increasing function in δ.2. For fixed δ, put F (n) = γ(δ). Since F ′(n) = δ∆/(1− (n− 1)δ)2 > 0,γ(δ), γ(c) and γ(δ; c) are increasing functions in n for fixed δ.3. If Sj(z − cj1) ≈ Sj(z − cj2), j = 1, · · · , n, then γ(c) ≈ 0. Further ifcjk ≈ 0, j = 1, · · · , n, k = 1, 2, then we can take δ1 ≈ δ ≈ δ2 besidesγ(c) ≈ 0.4. νc(Ec(δ, c;M)) is a decreasing function in δ.5. When qj is close to the neighbor ones, it will be difficult to find astep in the graph of (Gη(M ; c))(x) or we may take two peaks for onepeak in the graph of (G′η(M ; c))(x). Thus it is preferable that ∆ isnot small.6. In general, ∆′ is not small. Therefore, Sj(z − cj1) ≈ Sj(z − cj2), j =1, · · · , n, are necessary to γ(c) ≈ 0.4. Remark for applicationsTo estimate the number of sources, both γ(δ, c) and ρn(δ, c;M) are ex-pected to be small enough. By remarks 1, 2 and 5 in §3.4, since ∆ is expectedto be suitabley large, we have to take δ so small if we have many sources(i.e. n is large). This means, for large n, we can estimate the number ofsources only the case that there exist domains such that each Sj dominatesover the other sources. Except the trivial cases, it will be hard to estimatethe number of sources when we have many sources. Further, in general, wecan not ignore the time delay; that is, at leaset one cjk does not vanish. Butby remark 6 in §3.4, Sj(z − cj1) ≈ Sj(z − cj2), j = 1, · · · , n, are expected.As an example, we will consider the case that two observed signalsare given by (1). To transform the sources in the time-frequency domain,we will take an wavelet transformation. The continuous wavelet transformWψs(t, ω) of s ∈ L2(R) is defined byWψs(t, ω) = |ω|−1/2∫Rs(x)ψ(x− tω)dx,ON THE QUOTIENT FUNCTION EMPLOYED IN . . . 505where ψ is a wavelet function (see [1], for example). Thus, when we considerwavelet transforms of sources with time delay, we have to study the quotientfunction such asQ(t, ω) =∑nj=1 ajSj(t− cj1,1, ω)∑nj=1 bjSj(t− cj2,1, ω).This is the case that X = R2(t,ω) and cjk = (cjk,1, 0).For a numerical experiment, see [2] and [3] for example.References[1] R. Ashino, C. A. Berenstein, K. Fujita, A. Morimoto, M. Morimoto,D. Napoletani, and Y. Takei, Mathematical background for a methodon quotient signal decomposition. Applicable Analysis 86, No 5 (2007),577-609.[2] R. Ashino, T. Mandai, A. Morimoto, and F. Sasaki, Blind source sepa-ration using time-frequency analysis. Fields Institute Communications52 (2007), 401-414.[3] R. Ashino, T. Mandai, A. Morimoto, and F. Sasaki, Blind source sepa-ration of spatio-temporal mixed signals using time-frequency analysis.Applicable Analysis 88, No 3 (2009), 425-456.[4] R. Balan and J. Rosca, Statistical properties of STFT ratios for twochannel systems and applications to blind source separation. In: Pro-ceedings ICA 2000, 19–22 June 2000, Helsinki, Finland.[5] K. Fujita, On a method of blind source separation. In: Proc. of the FifthInternational conference on Information, 2009, 204-207.[6] K. Fujita, Remarks on a method of blind source separation. Information13, No 3(B) (2010), 829-831.[7] K. Fujita, Y. Takei, A. Morimoto, R. Ashino and M. Morimoto, Blindsource separation on a time-frequency space–mathematical background.In: IEICE Technical Report EA2005-12 (2005), 37-42 (In Japanese).[8] K. Fujita, Y. Takei, A. Morimoto and R. Ashino, Mathematical view ofa blind source separation on a time frequency space. Applied Mathe-matics and Computation187 (2007), 153-162.[9] A. Jourjine, S. Rickard and O. Yilmaz, Blind separation of disjoint or-thogonal signals: Demixing n sources from 2 mixtures. In: Proc. IEEE506 K. FujitaInternational Conf. on Acoustics, Speech and Signal Processing, 5–9June 2000, Istanbul, Turkey, IEEE Press, 2000.[10] D. Napoletani, C. A. Berenstein and P. S. Krishnaprasad, Quotient sig-nal decomposition and order estimation. Technical Research Report ofUniversity of Maryland, TR 2002-47, Institute for Systems Research,University of Maryland, College Park, MD, 2002; available online fromtechreports.isr.umd.edu/ARCHIVE/.Fuculty of Culture and EducationSaga University8408502, Saga – JAPANe-mail: keiko@cc.saga-u.ac.jp Received: GFTA, August 27-31, 2010

On the Quotient Function Employed in the Blind Source Separation Problem

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On the Quotient Function Employed in the Blind Source Separation Problem

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