SVD ( ) Linear Algebra for. A bit of repetition. Lecture: 8. Let s try the factorization. Is there a generalization? = Q2Λ2Q (spectral theorem!

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1 Liea Algeba fo Wieless Commuicatios Lectue: 8 Sigula Value Decompositio SVD Ove Edfos Depatmet of Electical ad Ifomatio echology Lud Uivesity it Ove Edfos A bit of epetitio A vey useful matix decompositio comes fom the Spectal heoem: EAL CASE A eal symmetic matix A ca be factoed ito QΛQ Q. he otoomal eigevectos of A ae i the othogoal matix Q ad the coespodig eigevalues i the diagoal matix Λ. COMPLEX CASE A emitia matix A ca be factoed ito UΛU. he otoomal eigevectos of A ae i the uitay matix U ad the coespodig eigevalues i the diagoal matix Λ. We have see that we, e.g., ca educe the complexity of chael estimatio i systems ad decouple othewise coupled systems liea systems. What a pity that it oly wos with symmetic o emitia matices, i.e., vey specific types of squae matices! Ove Edfos Is thee a geealizatio? Let s ty the factoizatio Would it be possible to eplace (the esticted A= QΛQ whee Q is othogoal ad Λ is diagoal, with somethig (moe geeal lie A= U ΣV whee U ad V ae still othogoal (but ot ecessaily the same ad Σ is still diagoal? his would give us essetially the same othogoal tasfom popety as we have used whe we had symmetic o emitia matices.... so, does this wo? Ove Edfos 3 Let s assume that a factoizatio extists. What would that imply? Fist loo at AA :... the at A A: A= UΣV ( Σ AA = UΣ V V Σ U = UΣΣ U ( AA= VΣ U UV Σ = VΣΣV = QΛQ (spectal theoem! = QΛQ (spectal theoem! his seem to wo oly if AA ad A A have some vey specific elatios betwee thei eigevalues Ove Edfos 4

2 Eigevalues of AA ad A A he sigula value decompositio Let s fid out the popeties of the eigevalues of AA ad A A. Do they have the same eigevalues? If x is a eigevecto of AA ad λ the coespodig eigevalue AA x = λx the, multiplyig by A fom the left, AAAx= λax shows us that A x is ad eigevecto of A A ad the same λ is its eigevalue. Ay othe useful popeties? YES! he above leads us to the followig theoem: Sigula Value Decompositio Ay M x N matix A ca be factoed ito A= UΣV he colums of U (M x M ae eigevectos of AA, ad the colums of V (N x N ae eigevectos of A A. he sigula values o the diagoal of Σ (M x N ae the squae oots of the ozeo eigevalues of both AA ad A A. Sice AA ad A A ae symmetic, the eigevalues ae eal. Note that t thee ae o estictios o A. It does t Sice the quadatic fom x A Ax = Ax 0, all eigevalues eve have to be squae. of A A (ad AA ae o-egative. his is a vey poweful decompositio... it wos o eveythig ad gives us the pactical (othogoal(diagoal(othogoal fom that simplifies may poblems! Ove Edfos Ove Edfos 6 he sigula value decompositio SVD of symmetic matices? About the stuctue: M x N Ae ofte soted σ σ... σ M x M M x N N x N σ v A= UΣV = u um σ v u to u spas C(A u + to u M spas N(A Implies that = a(a Sice the sigula values ae the squae-oots of the o-zeo eigevalues of both AA ad A A, thee ca be at most mi(m, of them, i.e., mi(m,. v to v spas C(A v + to v N spas N(A What about whe A = A? A = UΣV We ow that thee is a spectal factoizatio A = QΛQ of such a symmetic matix, ad we have Coclusio: he AA = QΛQ QΛ Q = QΛ Q sigula values of a symmetic matix eal eigevalues Squae oot of these diagoal ae the absolute elemets ae sigula values values of its ozeo eigevalues. Ca you fid a complete SVD fom the spectal factoizatio? I cetai applicatios, e.g. liea estimatio with stog coelatio, we have the popety that << mi(m, ad we ca use this to simplify calculatios Ove Edfos Ove Edfos 8

3 A MIMO system Capacity of MIMO-systems X Chael X M tasmit ateas M eceive ateas Ove Edfos Ove Edfos 0 A geeal (aow-bad model he geeal case with M X ateas ad M X ateas: y h, h, h, M x y h, h, h, M x y = = + = x+ ym h x M, hm, hm, M M M eceived vecto (M x Chael matix (M x M asmitted eceive vecto oise (M x vecto (M x It is ot tivial to figue out the capacity of this MIMO system, but the ectagula chael matix calls fo tyig a sigula value decompositio to obtai a somewhat simple system to aalyze Ove Edfos Capacity No fadig & AWGN [] Sigula value decompositio of the (fixed chael : y = x+ = UΣV x+ whee U (M x M ad V (M x M ae uitay matices ad Σ (M x M is a matix cotaiig the sigula values o its diagoal. Multiply by U fom left: U y = Σ V x+ U y x Oly otatios of y, x ad. y = Σx + All-zeo, exept diagoal Ove Edfos

4 Capacity No fadig & AWGN [] Capacity No fadig & AWGN [3] What have we obtaied? Paallel idepedet chaels: σ y = σ x+ 0 Numbe of sigula Shao s stadad case : σ x y x σ values = a(. (+ chaels with σ = 0 y Shao: he total capacity of paallel idepedet chaels is the sum of thei idividual capacities. C = log + ( + SN log ( + C = C = + SN Equal powe distibutio (chael ot ow at X: Costat dep. o e.g. X powe ad oise. log( log ( = = C = C = + ασ = + ασ his does t loo lie what we usually see about MIMO capacity Ove Edfos Ove Edfos 4 Capacity No fadig & AWGN [4] Capacity No fadig & AWGN [5] MIMO capacity is ofte see o the fom: C = log det ( IM + α... so, let s see if they ae the same! + ασ What we deived! C ( + ασ = log + g = = log det + = log det + ασ = ( I α ΣΣ = log det + α M ( M = log det UU + α UΣV VΣ U M I U I ΣΣ U = log det I ( M + α YES! Ove Edfos 5 CONCLUSION: Sigula values of aloe detemie the MIMO chael capacity (at a give SN! ( ( M C = log + ασ = log det I + α [bit/sec/z] = Nomalizatio: ρ - SN at each eceive bach ρ C = log det IM + M his elatio is also deived (i a diffet way i e.g G.J. Foschii ad M.J. Gas. O Limits of Wieless Commuicatios i a Fadig Eviomet whe Usig Multiple Ateas. Wieless Pesoal Commuicatios, o 6, pp , Ove Edfos 6

5 System model Simplified model ude ideal coditios (slow eough fadig ad sufficiet CP 0, 0, Chael estimatio i with pilot-symbol assisted modulatio x 0, y 0, x N, otal filte i the sigal path: = * * N, hsigal N, ( t hx ( t h( t hx ( t ( f = ( f * ( f * ( f y N, sigal X X systems have N betwee 64 (WLAN ad 89 (DigV. Give that subcaie is tasmitted at fequecy f the atteuatios become: =, sigal ( f Ove Edfos Ove Edfos 8 System model [cot.] We have eded d up with a matix model: y = Xh + whee y is the eceived vecto, X a diagoal matix with the tasmitted costellatio poits o its diagoal, h a vecto of chael atteuatios, ad a vecto of eceive oise. Chael estimatio What have we obtaied? hh ( hh +σ I U ( +σ I Λ Λ y Full NxN h y N- Diagoal NxN N- h matix poit matix poit multiplicatio. IFF multiplicatio. FF U Fo the pupose of chael estimatio, assume that all oes ae tasmitted, i.e., that X = I. We ow have a simplified model: All matices i calculatios l Same ume of ae squae, of equal size measuemets as y = h+ (ad symmetic. paametes to estimate! Futhe assume that the chael is zeo-mea ad has autocoelatio hh while the oise is i.i.d zeo mea complex Gaussia with autocoelatio = σ I. We also assume that h ad ae idepedet. N COMPLEXIY (opeatios Nlog N + N + Nlog N Nlog N N = Ove Edfos Ove Edfos 0

6 Chael estimatio (scatteed pilots all N subc caies Pilots o Fom ealie lectue system (X, Chael, X subcaie es s o all N Measuemet ta U ow dat Pilots scatteed i uow data system (X, Chael, X ts No o chael mea asueme Ove Edfos Chael estimatio (scatteed pilots Duig Lectue 7 we modelled the case with all pilots as a completely ow diagoal data matix X = I. We ow have x x 3 y = Xh + = h + x 4 If the uow data (x s ae zeo mea ad idepedet, we caot obtai ay ifomatio about the chael (h fom measuemets (y o those subcaies. We have to ely o the subcaies whee we have ow data (pilots Ove Edfos Chael estimatio (scatteed pilots o simplify the situatio, assume a ew (ediced size model whee we oly collect measuemets fom the P pilot subcaies. y = + h Measuemets Pilot matix Full chael eceive oise o the P pilot subcaies (P x [data ows emoved] (P x N vecto (N x vecto (P x LMMSE Scatteed Pilot Estimatio he liea estimato we ae looig fo is LMMSE = h whee A is a N x P matix o the fom A = hy Ay yy usig the coss coelatio hy betwee the N chael coefficiets h ad the P measuemets y ad the autooelatio yy of the P measuemets. Numbe of opeatios equied to pefom oe estimatio is: PN We have a ectagula matix ad the ice calculatios fom Lectue 7 have to be doe i a diffeet way! Smells lie a SVD! If we wat to educe the complexity of this estimato, we ca use the theoy of low-a estimatos. [Details ca be foud i boos about estimatio theoy.] y] Ove Edfos Ove Edfos 4

7 Chael est. low-a appoximatio By pefomig a SVD o the followig matix: / hy yy = UΣV... the best a-qq estimato is give by a a-qq appoximatio of the A matix: σ A A q = U σ V q / yy Ove Edfos 5 Chael est. low-a appoximatio σ h / LMMSE q = U V y yy σ q [... eep oly the q used colums of U ad V...] σ / = Uq Vq yy y σ q N x q q x q q x P P x P If the chael has stog coelatio the a q ca be made small. Compae to the NP equied without SVD. Combie to oe matix, q x P Numbe of opeatios equied to pefom oe estimatio: qp + qn = q(p+n Ove Edfos 6