Linear Algebra Issues in Wireless Communications

Transcription

1 Rome-Moscow school of Matrix Methods ad Applied Liear Algebra August 0 September 18, 016 Liear Algebra Issues i Wireless Commuicatios Russia Research Ceter [vladimir.lyashev@huawei.com]

2 About me ead of Iterferece Cacellatio Lab. at uawei Techologies, Russia Research Ceter, Wireless Research Divisio. Research Fellow at Souther Federal Uiversity, the Istitute of Sigal Processig ad Cotrol Systems Iteret of Iformatio Iteret of Thigs Iteret of Cotrol Page

3 Aim & Short Outlie Brief itroductio i the state-of-the-art problems of matrix calculatios i wireless commuicatio should allow participats to uderstad importace of mathematics ad mai problems here, which become part of our life. The itroductio is preseted by 3 parts: Itroductio i wireless commuicatio ad its relatio to liear algebra problems System model of wireless commuicatio Capacity of the system ad liear space extesio Defiitio of system matrix ad its properties discussio Outloo of matrix methods that are playig importat role i telecom. Matrix methods ad requiremets to them from the idustry Not well-posed problems i the sese of adamard (ill-posed problems) QR, LU, iterative methods ad requiremets to them from wireless systems Importat role of SVD i ultra-high rate commuicatios Belief propagatio approach i detectio problems Implemetatio issues Robustess & accuracy The simplest oe does t mea the fastest oe // free discussio Page 3

4 A istory [ 1948 ] Groudbreaig article by Claude E. Shao A mathematical theory of commuicatio itroducig the defiitio of capacity i commuicatio systems: The chael capacity is a measuremet of the maximum amout of iformatio that ca be trasmitted over a chael ad received with a egligible probability of error at the receiver. C B f log 1 SNR Iformatio is the resolutio of ucertaity. - Claude Elwood Shao Father of Iformatio Theory Page 4

5 Page 5 PDF: additive oise x y X Y P X Y P X P X Y Y P Y P Y ) ( )log ( ) ( ) ( ) ( )log ( ) ( X1 = +1; X = -1 Y1 = +1; Y = -1 P(X1) = 0.5; P(X) = 0.5 P(Y1 X1) = ; P(Y1 X) = ) P(Y X1) = ; P(Y X) = ) Error probability: 0.5%

6 Page 6 Sigal-to-Noise Ratio ; ) )( ( 0 0 x y P x x xx x x yy P x y 1 log 1 log log 1 x x y x y P B N P B N P B N P P W 0 pass bad 1 0 B W B B N B f

7 Detectio problem Page 7 X 1,,,3 Search costrait: Search strategy: X -? aswer ca be YES/NO bisectio method or iterval halvig or biary search or dichotomy method. As me Q#1 Q# Q#3 Q#4 Q#5

8 Resume 5 aswers YES/NO 5 bits of iformatio I P logm log1 artley's law: quatity of iformatio M, which is ecessary for detectio the specific value, is the base--logarithm of the umber of distict messages M that could be set. Page 8

9 Ope Problems i Wireless Commuicatio MULTI-USERS SYSTEMS (CDMA, SCMA, MU-MIMO, etc.) MULTI-ANTENNA SYSTEMS (massive-mimo, cell splittig/atea selectio, precodig/beamformig) Page 9

10 MIMO System Evolutio MORE spectrum efficiecy MORE atea elemets Page 10

11 Multiple Iput Multiple Output Commuicatios y x Page 11

12 Capacity of the system ad liear space extesio Page 1 P TX power of sigle trasmitter; N TX umber of trasmissio ateas; Q covariace matrix of trasmitted sigal. I curret product it is required matrix operatios for N RB (6 110) matrices with size 4x4 18x18 per 1ms or less I future N RB ca be exteded to 500, ad time scale ca be reduced 10 times!!!

13 What is chael matrix? Page 13

14 State-of-the-Art Problem To solve the matrix equatio as detectio problem Maximize Sigal-to-Noise Ratio for better performace User groupig for better throughput Page 14

15 Matrix applicatios i MIMO Eigevalue problems for beamformig ad iterferece cacellatio Reduced-ra sigal processig for regularizatio of iverse problems for atea combiig algorithms ad distributed MIMO/schedulig systems Low complexity algorithms for adaptive spatial filterig (dirty paper codig, STBC) System idetificatio (pilot cotamiatio problem i massive MIMO, type of traffic) Tesor Algebra for joit sigal processig (chael estimatio, multiuser detectio) Matrix volume maximizatio for MU-MIMO ad o-orthogoal access Polyomial series ad approximatio theory i itermodulatio compoets suppressio Page 15

16 Matrix Algebra for MIMO Problems Matrix Algebra is coveiet istrumet for problem formulatios i iformatio ad commuicatios systems ad to apply the cocepts ad algorithms of umerical liear algebra for optimal desig ad operatio of these systems Importat math. areas for MIMO Systems QR factorizatio ad least squares problems Coditioig ad stability issues i umerical liear algebra Numerical methods for solvig systems of liear equatios Eigevalue problems ad the computatio of the sigular value decompositio Aroldi algorithm Laczos iteratios CUR matrix decompositio Gradiets based approaches for o-liear problems Page 16

17 Vector Precodig The set of equatios that describe the MMSE solutio for vector precodig are: y Set x x ps p arg mi pˆ that R Thus, y U ΛV uu 1 y pˆ s Usig SVD, the matrix ca be diagoalized by orthogoal matrices U ad V x by trasmittig x xv U Λ Istead of x ad pre-multiply the receive sigal by vector U, the trasformed received sigal vector becomes: U Λ V x U Λx ~ y U y U ~ I TDD systems, the chael is the same o trasmitter ad receiver, but it is chagig i time ad robust precodig of the chael is challegeable problem i wireless comm. xv V Page 17

18 SVD is very importat operatio!!! Page 18

19 Eigevector calculatio algorithm Page 19

20 Tihoov Regularizatio i Iverse Problem Each least squares problem has to be regularized. I the liear case, we wat to solve miimizatio problem after regularizatio the solutio is x Page 0 Ax b Ax b ε Γx 1 A A Γ Γ A 1 A A I A b mi b ow to defie Tihoov matrix Г? Ax b ε x 5

21 Resume Typical size of the matrix is less elemets. For such matrix we eed fast algorithms for eigevector decompositio; matrix iversio (curret baselie is classical Cholesy decompositio algorithm) The questios are: 1.Ca we defie some less complexity algorithm for matrix iversio ad eigevector calculatio tha provided baselie algorithms?.do some fast algorithms (approaches) exist i moder liear algebra to compute such specific matrices? Page 1

22 omewor 1. Geerate Wishart-matrix A with followig parameters of distributio: Sigma = df = 8; Sigma = 0.1*oes(4) + 0.9*eye(4); A = wishrd(sigma,df)/df df = 8. Set up 000 samples of equatio: A ε x b ε, where x [ 1, 1]; (0, ), p is subject to A xˆ b p s, s. Page

23 omewor 3. Solve oisy equatio sample by sample ad defie probability of right solutio averaged over all samples. 1 p 1,1,1, // default vector T 1 4 p 1 (!) (1) 4. Repeat item #3 for p u - eigevector of matrix A, correspoded to the largest sigular value. 5. ow stochastic iformatio about additive oise ε ca be utilized for miimizatio of error probability?.. Page 3

24 omewor I. Geerate 000 samples of metrix A ad eep i memory for all umerical experimets. II. Set mappig vector p (two ways). III. Map oe-bit symbol (-1/+1) from 1x1 to 4x1: x = ps. IV. Compute b = Ax. V. Add gaussia oise (sigma is variatio parameter for aalysis): b = b +. VI. Solve oise equatio: Ax = b, that defie x as estimatio value. VII. Fid a way to defie s by ow p ad estimated x. VIII. Chec: how may s = s?... P error = 1 - <right s > / <umber of samples> IX. P error ca be defied as fuctio of deviatio of oise (sigma). Page 4