This paper analyzes applications of cumulant analysis in
speech processing. A special focus is made on different second-order
statistics. A dominant role is played by an integral representation for
cumulants by means of integrals involving cyclic products of kernels

Solé-Casals, Jordi

Zaiats, Vladimir

RIUVic

On Cumulant Techniques in Speech ProcessingVladimir Zaiats and Jordi Sole´-CasalsDigital Technologies Group, Universitat de Vic, c/. Sagrada Famı´lia, 7, 08500 Vic(Barcelona), Spain,{vladimir.zaiats,jordi.sole}@uvic.catAbstract. This paper analyzes applications of cumulant analysis inspeech processing. A special focus is made on different second-orderstatistics. A dominant role is played by an integral representation forcumulants by means of integrals involving cyclic products of kernels.Keywords: Cumulants, higher-order statistics, correlogram, speech enhancement1 IntroductionDifferent methods in speech recognition use linear and non-linear proceduresderived from the speech signal by matching the autocorrelation or the powerspectrum, [4, 6, 5]. Many of these methods perform well for clean speech, whiletheir performance decreases strongly if noise conditions mismatch for trainingand testing.We obtain a representation for cumulants of second-order statistics contain-ing a special type of integrals that involve cyclic products of kernels. Our tech-niques are based on [1–3, 7].2 Integrals Involving Cyclic Products of KernelsFor m ∈ N, define Nm := {1, . . . ,m}. Assume that (V,FV) is a measurable spaceand µ1, . . . , µm are σ-finite (real- or complex-valued) measures on (V,FV). Form ∈ N, m ≥ 2, consider the following integral:xI (K1, . . . ,Km; ϕ) (1):=∫· · ·∫Vm[m∏p=1Kp(vp, vp+1)]ϕ(v1, . . . , vm)µ1(dv1) . . . µm(dvm)where vm+1 := v1. Integral (1) will be called an integral involving a cyclic productof kernels (IICPK).We will denote:x∏p∈NmKp(vp, vp+1) :=m∏p=1Kp(vp, vp+1)with vm+1 := v1. This function will be called a cyclic product of kernelsK1, . . . ,Km.3 Cumulants of General Bilinear Forms of GaussianRandom VectorsSuppose that m ∈ N; nj,1, nj,2 ∈ N, j ∈ Nm, and writeXj,1 := (Xj,1(k), k ∈ Nnj,1), Xj,2 := (Xj,2(k), k ∈ Nnj,2), j ∈ Nm.Assume that Xj,1 and Xj,2, j ∈ Nm, are real-valued zero-mean random vectorsand consider the following bilinear forms:Uj :=nj,1,nj,2∑k,l=1aj(k, l)Xj,1(k)Xj,2(l), j ∈ Nm,wherenj,1,nj,2∑k,l=1:=nj,1∑k=1nj,2∑l=1.If we put(aj(k, l)) :=aj(1, 1) . . . aj(1, nj,2)... . . ....aj(nj,1, 1) . . . aj(nj,1, nj,2) , j ∈ Nm,then for any j ∈ NmUj =Xj,1 (aj(k, l))X⊤j,1= (Xj,1(1), . . . , Xj,1(nj,1))aj(1, 1) . . . aj(1, nj,2)... . . ....aj(nj,1, 1) . . . aj(nj,1, nj,2)Xj,2(1)...Xj,2(nj,2) .Consider the joint simple cumulant cum (U1, . . . , Um) of the random variablesU1, . . . , Um assuming that this cumulant exists. By general properties of thecumulants, we obtaincum (U1, . . . , Um)=n1,1,n1,2∑k1,1,k1,2=1· · ·nm,1,nm,2∑km,1,km,2=1 m∏j=1aj(kj,1, kj,2)× cum (Xj,1(kj,1)Xj,2(kj,2), j ∈ Nm) .Since any general bilinear form can be represented as a sum of diagonal bilinearforms, the following result holds.Theorem 1. Let m ∈ N; nj,1 = nj,2 = nj ∈ N, j ∈ Nm. Assume that (Xj,1,Xj,2, j ∈ Nm) is a jointly Gaussian family of zero-mean random variables andsuppose that for any j, ˜ ∈ Nm and any α, α˜ ∈ {1, 2} there exists a complex-valuedmeasure Mα,α˜j,˜ such thatEXj,α(k)X˜,α˜(k˜) =∫Dei(k−k˜)λMα,α˜j,˜ (dλ).Thencum (U1, . . . , Um)=∑l∈Lm(nj ,j∈Nm)∑(,α)∈{P,2}m−1∫· · ·∫Dm x∏p∈NmK(,α,l)p (vp − vp+1)× µ(,α,l)1 (dv1) . . . µ(,α,l)m (dvm),that is cum (U1, . . . , Um) is represented as a finite sum of integrals involvingcyclic products of kernels. Here,  := (j1, j2, . . . , jm), α := (α1, α2, . . . , αm),jm+1 = j1 = 1, αm+1 = α1 = 2, and the sum∑(,α) is extended to all((j2, . . . , jm), (α2, . . . , αm)) ∈ Perm{2, . . . ,m} × {1, 2}m−1. (2)The notation (,α) ∈ {P, 2}m−1 for fixed j1 = 1 and α1 = 2 is equivalent to (2).Here, we put Z|nj−1| := {−(nj − 1), . . . ,−1, 0, 1, . . . , nj − 1} for j ∈ Nm andLm(nj , j ∈ Nm) := Z|n1−1| × . . .× Z|nm−1|.4 ApplicationsWe apply the above obtained integral representations to some problems in speechrecognition. Let us consider a setting where sample correlograms and samplecross-correlograms of stationary time series appear.Let Y (t) := (Y1(t), Y2(t)), t ∈ Z, be a weak sense stationary zero-meanbidimensional vector-valued stochastic process with real-valued components whosematrix-valued autocovariance function is as follows:CY (t) :=(C11(t) C12(t)C21(t) C22(t)), t ∈ Z,and letFY (λ) :=(F11(λ) F12(λ)F21(λ) F22(λ)), λ ∈ [−π, π],be the matrix-valued spectral function of the vector-valued process Y (t), t ∈ Z.Let γ, δ ∈ {1, 2}. Consider the following random variables:Cˆγδ(τ ;N) :=N∑k=1bγδ(k; τ,N)Yγ(k + τ)Yδ(k), τ ∈ Z, N ∈ N,where bγδ(k; τ,N), k ∈ NN , τ ∈ Z, N ∈ N, are non random real-valued weights.It is often assumed thatN∑k=1bγδ(k; τ,N) = 1, τ ∈ Z, N ∈ N. (3)For example, let N ∈ N be given and letbγδ(k; τ,N) =1N, k ∈ NN , τ ∈ Z.Then (3) holds andCˆγδ(τ ;N) =1NN∑k=1Yγ(k + τ)Yδ(k), τ ∈ Z, N ∈ N. (4)The following sample correlograms are also often used in spectral analysis andspeech recognition as estimates of Cγδ(·), γ, δ ∈ {1, 2} :C˜γδ(τ ;N) =1NN−|τ |∑k=1Yγ(k + |τ |)Yδ(k), for |τ | < N ;0, for |τ | ≥ N,˜˜Cγδ(τ ;N) =1N − |τ |N−|τ |∑k=1Yγ(k + |τ |)Yδ(k), for |τ | < N ;0, for |τ | ≥ N.Let γ, δ ∈ {1, 2}, N ∈ N, m ∈ N, and τj ∈ Z, j ∈ Nm. Putcum(N)γδ (τ1, . . . , τm) := cum(Cˆγδ(τj ;N), j ∈ Nm);nj = N, j ∈ Nm; aj(k) = bγδ(k; τj , N), k ∈ NN , j ∈ Nm;Xj,1(k) = Yγ(k + τj), k ∈ NN , j ∈ Nm;Xj,2(k) = Yδ(k), k ∈ NN , j ∈ Nm.Under these conditions the results obtained in Section 3 can be applied to thecumulants. These results imply that the Gaussian component of the cumulantcum(N)γδ (τ1, . . . , τm) is represented as a finite sum of integrals involving cyclicproducts of kernels.Acknowledgements. We acknowledge the support of the Universitat de Vic un-der grant R0912. Some parts of this paper are based on joint research with V.Buldygin (National Technical University of Ukraine) and F. Utzet (UniversitatAuto`noma de Barcelona).References1. Avram, F.: Generalized Szego theorems and asymptotics of cumulants by graphicalmethods. Trans. Amer. Math. Society, 330, 637–649 (1992)2. Avram, F.; Taqqu, M.: On the generalized Brascamp-Lieb-Barthe inequality, aSzego˝ type limit theorem, and the asymptotic theory of random sums, integralsand quadratic forms. Manuscript. March 6, 20053. Grenander, U.; Szego˝, G. : Toeplitz forms and their applications. University ofCalifornia Press (1958)4. Makhoul, J.: Linear prediction: A tutorial review. Proc. IEEE, 63, 561–580 (1975)5. Nemer, E., Goubran, R., Mahmoud, S.: Speech enhancement using fourth-ordercumulants and optimum filters in the subband domain. Speech Comm., 36, 219–246 (2002)6. Paliwal, K. K., Sondhi, M. M.: Recognition of noisy speech using cumulant-basedlinear prediction analysis. Proc. ICASSP, 429–432 (1991)7. Sugrailis, D.: On multiple Poisson stochastic integrals and associated Markov semi-groups. Probab. Math. Statist., 3, 217–239 (1984)

On cumulant techniques in speech processing

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