A note on equiangular tight frames

Transcription

1 A ote o equiagular tight frames Thomas Strohmer Departmet of Mathematics, Uiversity of Califoria, Davis, CA 9566, USA Abstract We settle a cojecture of Joseph Rees about the existece a costructio of certai equiagular tight frames. Key wors: Equiagular tight frames, Sherma-Morriso-Woobury formula, coferece matrix Itrouctio A equiagular tight frame is a family of vectors {f k } k= i E (where E = R or C) that satisfies the coitios (0) f k 2 = for k =,...,, () f k, f l = c, for all k l a some costat c, (2) f, f k f k = f, for all f E m. (3) k= I fact, coitios (2) a (3) together imply that f k, f l =, for all k l, (4) () which is the smallest possible value for c for a set of equiagular uit-orm vectors i E, cf. (4; 0). Due to their rich theoretical properties a their umerous practical applicatios, equiagular tight frames are arguably the most importat class of aress: strohmer@math.ucavis.eu (Thomas Strohmer). T.S. ackowleges support from NSF DMS grat Preprit submitte to Elsevier 3 February 2008

2 fiite-imesioal frames, a they are the atural choice whe oe tries to combie the avatages of orthoormal bases with the cocept of reuacy provie by frames (0). Yet, espite their importace, we are far from havig a complete uerstaig about the existece of equiagular tight frames. Some results ca be fou i (2; 8; 4; 3; 3; 0; 6; ; 7; 9; ). A popular metho to costruct equiagular tight frames is base o coferece matrices (8; 3; 0). Before we procee to this costructio a the statemet of Rees cojecture, we ee to itrouce some otatio. For geeral backgrou o frames we refer to (2). Give a frame {f k } k= for E, the frame operator S is efie by Sf = f, f k f k. k= Note that the frame operator of a tight frame is a multiple of the ietity operator. The tight frame caoically associate to {f k } k= is {S 2 f k } k=. We will write (, )-ETF for a equiagular tight frame {f k } k= i E. Now, recall that a coferece matrix C has zeros alog its mai iagoal, ± as its other etries. a satisfies CC = ()I (I eotes the ietity matrix). Give a coferece matrix with = 2, oe ca costruct a (2, )-ETF {f k } k= via its Gram matrix R = { f l, f k } k,l=. If C is symmetric, oe computes R = C + I, if C is skew-symmetric (i.e., C = C T ) oe computes R = i C + I. Oe ca the extract the (2, )-ETF {f k } k= from R via a sigular value ecompositio, see (0). I (9), Joseph Rees cojecture that, give a (2, )-ETF {f k } k= associate with a skew-symmetric coferece matrix, oe ca always costruct a (2, )-ETF by removig a arbitrary frame elemet from {f k } k= a the computig the tight frame caoically associate with the remaiig frame elemets. This cojecture is supporte by umerical simulatios as well as by a proof by Rees for the special case whe satisfies the property 2 = p k = 3 mo 4, where p is a prime umber. Rees proof relies o Zauer s costructio of (2, )-ETFs a specific properties of fiite fiels a Gauss sums. I the followig we settle Rees s cojecture for geeral. Our proof uses oly elemetary liear algebra. Theorem. Let {f k } k= be a equiagular tight frame for C with = 2 a assume that f k, f l = ±i, for all k l. (5) Defie the frame {ϕ (l) j } j= := {f k } k l,k= a eote the frame operator asso- 2

3 ciate with {ϕ (l) j } j= by S l. Set g j = S 2 l ϕ (l) j, for j =,...,, the {g j } j= is a equiagular tight frame for C. Proof: Without loss of geerality we let l = a set ϕ k := ϕ () k = f k, for k =,..., (i.e., we remove the last elemet of the frame {f k } k=). We eote by F the matrix cotaiig the frame vectors f k, k =,..., as colums a similarly Φ a G are () matrices havig the ϕ k a g k as their colums, respectively. There hols G G = Φ (ΦΦ ) 2 (ΦΦ ) 2 Φ = Φ (ΦΦ ) Φ. (6) We apply the Sherma Morriso Woobury formula (5) a compute (ΦΦ ) = (F F f f ) = ( I f f ) = ( ) I + f f, (7) 2 where we have use that {f k } k= is a tight frame. We isert (7) ito (6) a obtai G G = ( ) Φ Φ + Φ f f Φ. Now cosier (G G) k,l for k, l =,..., ; k l: (G G) k,l = ( ) f l, f k + f l, f f, f k. (8) By assumptio f k, f l = ±i a therefore for all k, l =,..., with k l. Hece (G G) k,l = ( ± i ± ), (9) (G G) k,l = 2, (0) for all k, l =,..., ; k l, which completes the proof. Remark: Sice the off-iagoal etries of the Gram matrix of a (2, )-ETF associate with a skew-symmetric coferece matrix always satisfy coitio (5), Theorem. proves Rees cojecture. Corollary.2 A ecessary coitio for the Gram matrix of a (, )-ETF to satisfy f k, f l = ±i, for all k l, () () 3

4 is that = 2. Proof: We repeat the steps of the proof of Theorem. for arbitrary N with >. Deotig α = /, we have f k, f l = ±i (α ), for all k l. (2) (α ) The right-ha sie of equatio (0) ow becomes α(α ) α + 2. (3) α Equatig (3) with (4) a solvig for α gives as oly feasible solutio α = 2. Corollary.3 If {f k } k= is a real-value (, )-ETF, the the caoical tight frame associate with the frame obtaie by removig a arbitrary elemet from {f k } k= ca ever be equiagular, except for the trivial case = +. Proof: Let R eote the Gram matrix of a real-value (, )-ETF. We claim that the etries R k,l, k l caot all have the same sig uless = +. To see this we first recall that R has eigevalues that are equal to / a eigevalues equal to 0, cf. (0). Now assume that sig(r k,l ) = for all k l. I this case R is a circulat matrix a its eigevalues are give by the Discrete Fourier Trasform ˆr of the first colum r of R. Sice r = [, c, c,..., c] T, where c = ( )/( ), it follows that ˆr = [ c + ( + c)/, ( + c)/, ( + c)/,..., ( + c)/ ] T. Clearly, ˆr ca have at most its first etry equal to zero, a this ca happe oly if = +. I case sig(r k,l ) = for all k l, ˆr woul have oly strictly positive etries, cotraictig the fact that R must have > 0 eigevalues equal to zero. We ow repeat the steps of the proof of Theorem. for real-value frames, a with = α for some α > such that N. Equatio (9) becomes (G G) k,l = ( α ± ± α ). (4) From above we kow that the off-iagoal etries of G G caot all have the same sig, except if = +, which leas to the trivial case G G = I. Thus, 4

5 i orer to have a equiagular frame for > + we must have α + α = (5) which is ot possible. This completes the proof. Refereces [] D.M. Appleby. SIC-POVMs a the extee Cliffor group. Math. Phys., 46:05207, [2] O. Christese. A itrouctio to frames a Riesz bases. Birkhäuser, Bosto, [3] J.H. Coway, R.H. Hari, a N.J.A. Sloae. Packig lies, plaes, etc.: packigs i Grassmaia spaces. Experimet. Math., 5(2):39 59, 996. [4] P. Delsarte, J. M. Goethals, a J. J. Seiel. Bous for systems of lies a Jacobi poyomials. Philips Res. Repts, 30(3):9 05, 975. Issue i hoour of C.J. Bouwkamp. [5] G.H. Golub a C.F. va Loa. Matrix Computatios. Johs Hopkis, Baltimore, thir eitio, 996. [6] R.B. Holmes a V.I. Paulse. Optimal frames for erasures. Liear Algebra Appl., 377:3 5, [7] D. Kalra. Complex equiagular cyclic frames a erasures. Liear Algebra Appl., 49(): , [8] P. W. H. Lemmes a J. J. Seiel. Equiagular lies. J. Algebra, 24:494 52, 973. [9] J. Rees. Equiagular tight frames from Paley touramets. Liear Algebra Appl., 426(2-3):497 50, [0] T. Strohmer a R. Heath. Grassmaia frames with applicatios to coig a commuicatios. Appl. Comp. Harm. Aal., 4(3): , [] M. Sustik, J.A. Tropp, I. Dhillo, a R.W. Heath Jr. O the existece of equiagular tight frames. Liear Algebra Appl., 426(2 3):69 635, [2] J. H. va Lit a J. J. Seiel. Equilateral poit sets i elliptic geometry. Neerl. Aka. Wetesch. Proc. Ser. A 69=Iag. Math., 28: , 966. [3] G. Zauer. Quatum Desigs Fouatios of a No-Commutative Theory of Desigs. PhD thesis, Uiversity of Viea,