Orthogonality, orthogonalization, least squares
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1 ier Alger for Wireless Commuictios ecture: 3 Orthogolit, orthogoliztio, lest squres Ier products d Cosies he gle etee o-zero vectors d is cosθθ he l of Cosies: + cosθ If the gle etee to vectors is π/ (90º), e s tht the re orthogol. his mes θ Ove Edfors Deprtmet of Electricl d Iformtio echolog ud Uiversit it cos π or Ove Edfors Ove Edfors Proectio oto lie Cuch-Schrz-Buios iequlit Proectio of vector oto lie i the directio of : p α here α is chose so tht the lie from to p is the shortest oe. θ p From the formul for proectio oto lie, e c derive the most importt t iequlit i mthemtics ti... the Cuch-Schrz-Buios iequlit. It is ovious tht squred legth of the differece vector e p must e o-egtive: p 0 θ p his results i tht - p should e orthogol (perpediculr) to the lie (vector ). he proectio s mtri multiplictio (proectio mtri): p P P is r proectio mtri. his leds to or ( ) Ove Edfors Ove Edfors 4
2 Proectio oto suspce Proectio oto suspce (cot.) Proectio oto lie i the directio of the vector s performed r proectio mtri: P Proectio oto suspce sped the mtri A could e somethig similr, lie: et s fid out! AA P A A A A AA C t divide mtri Ove Edfors 5 We t to fid the closest poit/vector p i the colum spce C(A) ofthemtria A to poit/vector. he vectors p i C(A) c e ritte s p A he distce etee d p is p A hich is miimized if p is orthogol to C(A), or equivletl to ll the colums of A. his gives the lest squres solutio ( ) ( ) 0 A p A AS or I sttistics, these re AA S A clled the orml equtios. p C A Ove Edfors 6 Proectio oto suspce (cot.) Mtrices ith orthoorml colums Frome these orml equtios e c, if A A is ivertile, clculte the proectio p solvig for S : S A A A Iserig this i the epressio for p e get: p AS A A A A Mtri proectig oto the colum spce of A. Sme s our guess! Whe does this hppe? Whe the colums of A re lierl idepedet! he vectors q,..., q re orthoorml if qq i if 0 if i i (ormliztio) (orthogolit) For rectgulr (or squre) mtri Q ith orthoorml colms e hve QQ I hich implies tht it preserves the legth of vectors ccordig to Q Q Q d preserves ier products d gles ccordig to Q Q Q Q We c see them s rottio mtrices Ove Edfors Ove Edfors 8
3 Orthogol mtrices Crete orthogol vectors? A squre mtri ith orthoorml colums is sid to e orthogol mtri. For orthogol mtri Q e hve, i dditio to the previous, the folloig properties: Mtrices ith orthoorml colums come i hd i m situtios. heir computtiol properties re ttrctive ti... d Q Q orthoorml colums lso implies orthoorml ros. Coserves vector legth: (umericl stilit) Proectios ecome simple: (fster solutios of certi prolems) Q p QQQ Q QQ et s see ho e prepre set of o-orthogol vectors to oti orthoorml set Ove Edfors Ove Edfors 0 rm-schmidt orthogoliztio QR fctoriztio et,,,...,, e lierl idepedet M-vectors. Froom these, step-ise crete set of orthoorml vectors q,,,..., sutrctig from t step the compoets i the directios of lred settled vectors q, q,..., q - : d ormlizig the legth to uit: r q q q r r egth of s compoet log q. he orthoorml vectors q,,,...,, o form orthogol se for the spce sped the origil vectors,,,...,. he rm-schmidt orthogoliztio requires M opertios. he rm-schmidth orthogoliztio produces fctoriztio of mtri A ith colums,,,...,, i the folloig form: A M q q q q q q q q QR O M q Orthoorml colums Right (upper) trigulr M d divertile Ove Edfors Ove Edfors
4 QR fctoriztio (cot.) Emple: MIMO d QR () he QR fctoriztio simplifies the lest squres prolem A, for hich e hve the orml equtios: Usig A QR e get AA S A QR QR QR RQQR S S I RQ Sice R is right (upper) trigulr, this RR S RQ oe is esil solved c-sustitutio R Q S Ove Edfors 3 Assume tht e hve trsmissio sstem, here vector is trsmitted through h MIMO chel d, resultig i received vector + Further, ssume tht the the chel c e QR fctorized s QR Mig the sustitutio e get QR+ Usig the fct tht Q Q I, e come to the coclusio tht receiver side processig of Q the e get MIMO-sstem: z Q QR+ R+ Q Does ot chge sttisticl properties of. With resultig trigulr MIMO chel mtri detectio successive ccelltio of iterferece is simplified i the receiver i theor Ove Edfors 4 Q z Emple: MIMO d QR () Assume tht e hve trsmissio sstem, here vector is trsmitted through MIMO chel d dditive oise is dded, d the resultig received vector is + Further, ssume tht the (hermiti trspose) of the chel c e QR fctorized s QR R Q Mig the sustitutio e get R Q + Usig the fct tht Q Q I, e come to the coclusio tht precodig i the trsmitter Q the e get MIMO-sstem: R Q Q+ R + e trigulr chel mtri With resultig trigulr MIMO chel mtri successive ccelltio of iterferece lred o the trsmitter side is possile - t lest i theor Ove Edfors 5 Q Some importt orthogol mtrices Ove Edfors 6
5 -poit discrete Fourier trsform he o-ormlized versio (most commo) F F M M O M M M O M ( ) he orthogol versio ith ormlized colum vectors M M O M M M O M ( ) ( π / ) ep / Ove Edfors 7 -poit discrete Fourier trsform If the umer of poits i the discrete fourier trsmform is poer of, the mtri multiplictio (the trformtio) c e performed i log opertios isted of the orml Ove Edfors 8 -poit (Wlsh-)dmrd trsform -poit (Wlsh-)dmrd trsform he (Wlsh-)dmrd d trsform mtri c e recursivel defied d s: Its iverse is [ ] Applictios i sigl processig, dt compressio d orthogol spredig i ireless commuictio sstems. Emple: Ove Edfors Ove Edfors 0
6 -poit r trsform -poit r trsform he r trsform mtri c e recursivel defied d s: Its iverse is [ ] [ ] [ ] Belogs to the clss of discrete velet trsforms d hs pplictios i sigl processig, time series lsis d dt compressio. Emple: Ove Edfors Ove Edfors
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