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=HRxHT−HRxdHT,(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.