# Two important particular cases of 32 are immediate m

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Two important particular cases of (3.2) are immediate.M=1.In this situation,z1(k)=z(k)is a scalar andHsimplifies to an FIR filterhTof lengthL.This case was well studied inChap. 2.M=L.In this situation,zL(k)=z(k)is a vector of lengthLandH=HSis asquare matrix of sizeL×L.This scenario has been widely covered in  andin many other papers. We will get back to this case a bit later in this chapter.By definition, our desired signal is the vectorxM(k).The filtered speech,xMf(k),depends onx(k)but our desired signal after noise reduction should explicitly dependsonxM(k).Therefore, we need to extractxM(k)fromx(k).For that, we need todecomposex(k)into two orthogonal components: one that is correlated with (or is alinear transformation of) the desired signalxM(k)and the other one that is orthogonaltoxM(k)and, hence, will be considered as the interference signal. Specifically, thevectorx(k)is decomposed into the following form:x(k)=RxxMR1xMxM(k)+xi(k)=xd(k)+xi(k),(3.7)wherexd(k)=RxxMR1xMxM(k)=ŴxxMxM(k)(3.8)is a linear transformation of the desired signal,RxM=EbracketleftbigxM(k)xMT(k)bracketrightbigis thecorrelation matrix (of sizeM×M) ofxM(k),RxxM=Ebracketleftbigx(k)xMT(k)bracketrightbigis the cross-correlation matrix (of sizeL×M) betweenx(k)andxM(k),ŴxxM=RxxMR1xM,andxi(k)=x(k)xd(k)(3.9)
3.1 Linear Filtering with a Rectangular Matrix25is the interference signal. It is easy to see thatxd(k)andxi(k)are orthogonal, i.e.,Ebracketleftbigxd(k)xTi(k)bracketrightbig=0L×L.(3.10)For the particular caseM=L,we haveŴxx=IL,which is the identity matrix(of sizeL×L), andxd(k)coincides withx(k),which obviously makes sense. ForM=1,Ŵxx1simplifies to the normalized correlation vector (seeChap. 2)ρxx=E[x(k)x(k)]Ebracketleftbigx2(k)bracketrightbig.(3.11)Substituting (3.7) into (3.2), we getzM(k)=H[xd(k)+xi(k)+v(k)]=xMfd(k)+xMri(k)+vMrn(k),(3.12)wherexMfd(k)=Hxd(k)(3.13)is the filtered desired signal,xMri(k)=Hxi(k)(3.14)is the residual interference, andvMrn(k)=Hv(k),again, represents the residualnoise. It can be checked that the three termsxMfd(k),xMri(k),andvMrn(k)are mutuallyorthogonal. Therefore, the correlation matrix ofzM(k)isRzM=EbracketleftbigzM(k)zMT(k)bracketrightbig=RxMfd+RxMri+RvMrn,(3.15)whereRxMfd=HRxdHT,(3.16)RxMri=HRxiHT=HRxHTHRxdHT,(3.17)RvMrn=HRvHT,(3.18)Rxd=ŴxxMRxMŴTxxMis the correlation matrix (whose rank is equal toM) ofxd(k),andRxi=Ebracketleftbigxi(k)xTi(k)bracketrightbigis the correlation matrix ofxi(k).The correlation matrixofzM(k)is helpful in defining meaningful performance measures.
263Single-Channel Filtering Matrix3.2 Joint DiagonalizationBy exploiting the decomposition ofx(k),we can decompose the correlation matrixofy(k)asRy=Rxd+Rin=ŴxxMRxMŴTxxM+Rin,(3.19)whereRin=Rxi+Rv(3.20)is the interference-plus-noise correlation matrix. It is interesting to observe from(3.19
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