Introduction. Linear Algebra for. Lectures. Lecture: 1. Course material. Home assignments. Preliminary plan. One set of problems with each lecture.

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1 iner Algebr for Wireless Communictions ecture: Introduction ectures Preliminry pln About one lecture every nd week wo lectures before -ms (this nd net week) Detiled schedule Web pges here A detiled schedule is vilble on these pges, but it is subect to chnges Ove Edfors Deprtment of Electroscience und University it 9--9 Ove Edfors 9--9 Ove Edfors Course mteril here is no single tetbook in this course! he course slides constitute the contents of the course, while the mteril origintes from mny sources, e.g.: If you re buying one, I recommend dthi this one! G. Strng. iner Algebr nd Its Applictions. (test) Fourth edition: Brooks/Cole G.H. Golub. & C.F. vn on. tri Computtions. Johns Hopkins.. Kilth. iner Systems (Appendi). Prentice Hll. G. Strng. Introduction to Applied themtics. Wellesley Cmbridge Press. K. Ogt. Discrete-time Control Systems. Prentice-Hll... Schrf. Sttisticl Signl Processing. Addison Wesley. S.. Ky. Fundmentls of Sttisticl Signl Processing (Est. heory / Det. heory). Prentice Hll... Cover & J.A. homs. Elements of Informtion heory. Wiley. Home ssignments One set of problems with ech lecture. o pss the course, solutions must be hnded in before the net lecture, nd bout 8% correct (in totl over the whole course) Ove Edfors Ove Edfors 4

2 Why should you bother? Why should you bother? Becuse you find things like this in ournl ppers on wireless communictions:... or stuff like this: hese eples re from (conf. pper): uc Deneire nd Dirk.. Slock. A DEERIISIC SCHUR EHOD FOR UICHAE BID IDEIFICAIO. nd IEEEWorkshop on Signl Processing Advnces in Wireless Communictions, y 9-, Annpolis, D, USA, pp hese eples re from: Jeong Geun Kim & rwn. Krunz. Bndwidth lloction in wireless networks with gurnteed pcket-loss performnce. IEEE/AC rnsctions on etworking, vol 8, no 3, June 9--9 Ove Edfors Ove Edfors 6 Why should you bother? Simple emples: Pcket trnsmission over FIR chnnel... or stuff like this: trnsmitted dt points k FIR chnnel of length h k oise n k y k Received signl hese eples re from: O. Edfors et l. OFD chnnel estimtion by singulr vlue decomposition. IEEE rnsctions on Communictions, vol. 46, no. 7, pp , July Ove Edfors 7 ( ) y h + n h + n k k k n kn k n Over the interesting intervl between k nd k + - where there re contributions from the dt to the received signl, we hve n equivlent mtri model: h h y n h O y H+ n h h + y + n + O h 9--9 Ove Edfors 8

3 Simple emples: Pcket trnsmission over FIR chnnel Simple emples: ultiple trnsmit nd receive ntenns h h y n h O y H+ n + h h y + n + O h Some fundmentl problems: Known dt points re clled pilots. If the chnnel H nd the noise properties re known, we my wnt to detect the dt points from the received signl y. If the dt points nd noise properties re known, we my wnt to estimte the chnnel H (under the restrictions given by its structure). If the dt points re known, we my wnt to estimte both the chnnel H (under the restrictions given by its structure) nd the noise properties. X diversity (ISO):,, + y h h n RX diversity (SIO): y h, n + y h, n X&RX diversity (IO): y h, h, n y h, h +, n 9--9 Ove Edfors Ove Edfors Simple emples: ultiple trnsmit nd receive ntenns he generl cse with X ntenns nd R RX ntenns: y h, h, h, n y h, h, h, n y + + O H n y h R R, hr, hr, n Some fundmentl problems: - How do we model the chnnel mtri H? - How do we model the noise (interference) n? - How much dt cn we trnsmit through such chnnel? 9--9 Ove Edfors A few words bout vectors mtrices nd some opertions 9--9 Ove Edfors

4 Bsic norms Bsic mtri opertions Euclidin norm of n vector k Frobenius norm of n mtri A A F i k i, he Euclidin norm of vector is specil cse of the Frobenius norm of mtrices. tri multipliction is ASSOCIAIVE: ( ) ( ) A BC tri opertions re DISRIBUIVE: ( ) ( ) AB C A B + C AB + AC B+ C D BD+ CD tri multipliction is O COUAIVE: (usully) AB AB 9--9 Ove Edfors Ove Edfors 4 tri inverse tri trnspose AmtriA A is sid to be invertible if there eists mtri B such tht AB BA I Both must be fulfilled! here eists t most one such mtri B, it is clled the inverse of A nd is denoted A - : A A AA I,,,,,,,,, A O O,,,,,,,,, A he product of two invertible mtrices is invertible nd is given by: ( ) AB B A ote chnge of order! Properties ( ) AB B A ( ) A ( A ) 9--9 Ove Edfors Ove Edfors 6

5 tri Hermitin trnspose,,,,,, A O,,, H * A ( A ) ( A ) * denotes comple conugte of * * * *,,, * * *,,, O * * *,,, Inner product of vectors he inner product of two vectors nd y is y Rel cse [ ] y y y Comple (nd rel) cse y y H * * * y y he Euclidin norm cn be epressed using this inner product: Properties ( ) H H H AB B A ( ) H H A ( A ) H * * * i 9--9 Ove Edfors Ove Edfors 8 Hdmrd (k Shur) product Hdmrd (k Shur) product (cont.),,, b, b, b,,,, b, b, b, A B O O b, b, b,,,,, b,, b,, b,,b,,b,, b, A B O, b,, b,, b, Properties A B B A ( ) A B B A 9--9 Ove Edfors Ove Edfors

6 Kronecker product Kronecker product (cont.),,, b, b, b, Q,,, b, b, b, Q A B O O b P, bp, bp,,,, Q PQ, B, B, B,, B B, B A B O, B, B, B (P)(Q)(Q) Properties ( A+ B) C A C+ B C ( αa) B A ( αb) α( A B) ( A B) C A ( B C) ( A B)( C D) AC BD ( A B) A B 9--9 Ove Edfors 9--9 Ove Edfors tri trce Determinnt he trce of n mtri A is defined s tr ( A ), ii i (sum of digonl elements) he determinnt of n mtri A cn be interpreted s the volume of prllelepiped in R where the edges come from the columns (or rows) of A. z det A this volume Properties tr ( A ) tr ( A ) ( A+ B) ( A) + ( B) tr ( α A ) α tr ( A ) tr tr tr,,,3 A,,,3 3, 3, 3,3 he determinnt chnges sign depending on the order of the columns!,, 3,,3,3 3,3 y,, 3, 9--9 Ove Edfors Ove Edfors 4

7 Determinnt (cont.) Cofctor epnsion of determinnt he determinnt cn lso be interpreted s the chnge of volume when liner trnsformtion y A is pplied to body of certin volume. y A Volume V Volume V he determinnt of A cn be computed by epnding it in the cofctors of the i:th row: det A A + A + K+ A A i, i, i, i, i, i, i, i, where the cofctor A i, is the determinnt of the minor i, with the correct sign: ( ) i+ A i, det i, he minor i, of A is the mtri formed by removing row i nd column of A. V V det A his cn be used to recursively clculte the determinnt of ny mtri. After - steps we rrive t the sclr cse. his lso works long column Ove Edfors Ove Edfors 6 Crmer s rule Eigenvlues nd eigenvectors he :th component of is det B det A where A b,, b, +, B,, b, +, he vector b replces the :th column of A. he fundmentl eqution for the determintion of eigenvlues nd eigenvectors corresponding to the mtri A is A λ his eqution is nonliner, since it contins the product of two unknowns nd λ. Rewriting it s ( λ ) A I we see tht the number λ will only be n eigenvlue of A with corresponding non-zero eigenvector if nd only if ( A λi) det which is the chrcteristic eqution for the mtri A Ove Edfors Ove Edfors 8

8 Eigenvlues nd eigenvectors (cont.) Ech of the following conditions is necessry nd sufficient for the number λ to be n eigenvlue of A:. here is nonzero vector such tht A λ.. he mtri A-λI is singulr. 3. det(a-λi). he sum of the eigenvlues of A equls the sum of the digonl elements of A: λn n, n A n n tr ( ) tri types nd some of their properties he product of the eigenvlues of A equls the determinnt of A: n λ det n ( A) 9--9 Ove Edfors Ove Edfors 3 Digonl mtrices ringulr mtrices Identity I O Its own inverse All eigenvlues Determinnt Upper tringulr * * * * * * * U * * * Inverse upper tringulr Product U U upper tringulr Eigenvlues re on the digonl Determinnt product of dig. elements Digonl d D O d n Inverse digonl mtri Eigenvlues re on digonl Determinnt t product of digonl elements ower tringulr * Inverse lower tringulr * * * * * * * * * Product lower tringulr Eigenvlues re on the digonl Determinnt product of dig. elements 9--9 Ove Edfors Ove Edfors 3

9 Symmetric/Hermitin mtrices oeplitz nd circulnt mtrices Symmetric Skew-symmetric Hermitin Skew-Hermitin A A AA A A H Inverse is symmetric Eigenvlues re rel Inverse is skew-symmetric Digonl elements re zero Eigenvlues re imginry (or zero) Inverse is Hermitin Eigenvlues re rel Inverse is skew-hermitin H A A Digonl elements re zero Eigenvlues re imginry (or zero) A oeplitz mtri is squre mtri where ech decending digonl from left to right is constnt (elements only depend on the row/col inde difference i - ): ( ) O O O O A O O O O O O O If, in ddition, -k -k the mtri is clled circulnt mtri. Hs only - degrees of freedom Efficient numericl lgorithms oeplitz mtrices commute 9--9 Ove Edfors Ove Edfors 34 Hnkel mtrices A Hnkel mtri is squre mtri where ech skew-digonl is constnt (elements only depend on the row/col inde sum i + ): 3 O O 3 4 O O A O O O O 4 O O O Hs only - degrees of freedom Upside-down oeplitz Symmetric Efficient numericl lgorithms Gussin elimintion - fundmentl lgorithm If, in ddition, +k k the mtri is circulnt Ove Edfors Ove Edfors 36

10 Gussin elimintion Stndrd Gussin elimintion u 4 w w 7 Row Row Row Row 3 Row 3 + Row u w 4 3 w 8 Row 3 Row Row u u w 4 w 4 w 7 4 w Ove Edfors 37 he three elementry row opertions cn be epressed s series of lower tringulr mtri multiplictions Row Row Row E, Row3 Row3+ Row E3, Row3 Row3+ 3 Row E3, Ove Edfors 38 Applying the elemntry Guss trnsformtion mtrices to our coefficient mtri A 4 gives E3,E3,E, A he ppliction order of these mtrices is importnt! Wht s so specil with elemntry Guss trnsformtion mtrices? hey hve the following structure: E i, O e i, O O Ei, O O e i, O... lwys invertible nd inverse is very simple to clculte! 9--9 Ove Edfors Ove Edfors 4

11 et s tke closer look t our Gussin elimintion emple: Product of lower tringulr mtrices is lower tringulr ultiply by inverse from the left! I We hve demonstrted tht this A cn be decomposed into lower- nd n uppertringulr mtri. Inverse of lower tringulr is lower tringulr A U his is n U decomposition of A Ove Edfors Ove Edfors 4 he DU decomposition Definition: Property he pivot elements in Gussin elimintion re the digonl elements we use to eliminte the sub-digonl elements. If mtri cn be U decomposed,the pivot elements re those on the digonl of U. A mtri A cn be decomposed s the product U, of lower-tringulr mtri nd n upper-tringulr mtri U, s long s long s no pivot elements re zero Ove Edfors 43 he U decomposition of mtri A is slightly unsymmetric in the sense tht hs ones on its digonl, wheres U does not (in generl). u, u, u, l u, O, O A U O O O u, n l, l, u, A more symmetric version is the DU decomposition d % %,, u u l, d O A DU O O O O O u%, l, l, d where we hve (from the U decomposition) d u u% u / u u / d nd, i, i, i, i i, i 9--9 Ove Edfors 44

12 he DU decomposition (cont.) Property If mtri A cn be DU decomposed, then the three mtrices, D nd U re unique! Property If mtri A is symmetric nd cn be DU decomposed, then the upper trigulr mtri U is trnspose of the lower tringulr mtri nd A D 9--9 Ove Edfors 45 Will Gussin elimintion lwys work? If pivot element is zero, we re in more or less truble! ess trouble: Pi t l t b t th i ore trouble: 3 4 Pivot element zero, but there is non-zero element further down. Cn be fied by row ehnge! Pivot element zero nd ll elements below it re zero. his mtri is singulr nd it cn t be fied by row ehnge! 9--9 Ove Edfors 46 Row echnges re performed by permuttion mtrices. o echnge rows nd k multiply py by O from (,) to (,k) Is its own inverse Pk, O from (k,k) to (k,) O Cn row permuttions sve our U/DU decomposition? Assume tht t we need the following elementry Guss trnsformtions ti nd row echnges to complete the Gussin elimintion: Upper tringulr. E P E E A U 3,,3 3,, o longer lower tringulr, due to the permuttion mtri P,3. his will not led to U fctoriztion, since the inverse of E 3, P,3 E 3, E, cnnot be lower tringulr. A prtil solution: ke the necessry row echnges before the Gussin elimintion Ove Edfors Ove Edfors 48

13 In the non-singulr cse, there is permuttion mtri P tht reorders the rows of A, so tht PA dmits decomosition with non-zero pivots: PA U his permuttion P mtri is the product of the permuttion mtrices P,k required to complete the Gussin elimintion of A. In theory, n echnge of rows is only necessry when we encounter zero pivot element. In prctice, we obtin better numericl results if we lso mke row echnges when pivot elements re close to zero Ove Edfors Ove Edfors 5