Construction of Equiangular Signatures for Synchronous CDMA Systems

Transcription

1 Construction of Equiangular Signatures for Synchronous CDMA Systems Robert W Heath Jr Dept of Elect an Comp Engr 1 University Station C0803 Austin, TX USA rheath@eceutexaseu Joel A Tropp Inst for Comp Engr an Sci (ICES) 1 University Station C0200 Austin, TX jtropp@icesutexaseu Inerjit S Dhillon Dept of Comp Sci 1 University Station C0500 Austin, TX inerjit@csutexaseu Thomas Strohmer Dept of Mathematics University of California, Davis Davis, CA USA strohmer@mathucaviseu Abstract Welch boun equality (WBE) signature sequences maximize the uplink sum capacity in irect-sprea synchronous coe ivision multiple access (CDMA) systems WBE sequences have a nice interference invariance property that typically hols only when the system is fully loae an the signature set must be reesigne an reassigne as the number of active users changes to maintain this property An aitional equiangular constraint on the signature set, however, maintains interference invariance Fining such signatures requires imposing equiangular sie constraints on an inverse eigenvalue problem This paper presents an alternating projection algorithm that can esign WBE sequences that satisfy equiangular sie constraints The propose algorithm can be use to fin Grassmannian frames as well as equiangular tight frames Though one projection is onto a close but non convex set, it is shown that this algorithm converges to a fixe point, an these fixe points are partially characterize I INTRODUCTION Signature sequences that maximize the sum capacity in the uplink of irect-sprea synchronous coe ivision multiple access (CDMA) systems in Gaussian noise are known to satisfy Welch s boun on the total square correlation with equality [1] These sequences, known as Welch boun equality (WBE) signature sequences, are etermine by the number of users an the imensionality of the signature space They have the interesting interference invariance property in that each signature sees exactly the same interference power Thus the interference experience by a user is inepenent of the signature assigne to that user Unfortunately, when the number of active users changes, the signatures must generally be recompute an reassigne to maintain the interference invariance [2] Recently a class of signatures, known as Grassmannian signatures, were introuce that satisfy interference invariance even when subsets of the available users are active [3] This signature construction is intimately relate to the problem of sphere packing in the Grassmann manifol, in this case oneimensional subspaces (lines), an more specifically to the construction of Grassmannian tight frames [4] The interference invariance properties comes from the fact that because Grassmannian signatures satisfy Welch s lower boun on the maximum correlation with equality, they are equiangular (the correlation is the same for all istinct signature pairs) an maximally space with the smallest possible inner prouct The equiangular property provies interference invariance Unfortunately, signatures that are both equiangular an maximally space are quite rare Some explicit constructions are available in the articles [3], [4], [5] Signatures erive from goo packings in the Grassman manifol, even the best line packings tabulate by Sloane [6], o not generally satisfy the WBE property when they are not equiangular In cases where such signatures o not exist, we woul be satisfie with a WBE whose constituent signatures are close to equiangular Recently propose numerical algorithms for fining WBEs (eg, [7], [8], [9], [10]), however, o not easily incorporate equiangular sie constraints In this paper, we present an algorithm for fining Welch boun equality signature sequences that are exactly (or nearly) equiangular Our approach buils on our recently propose iterative algorithm for constructing CDMA signature sequences [11], which has also been use to fin signatures satisfying peak-to-average ratio constraints [12] The iea is to alternately solve two matrix nearness problems, one that fins the closest signature set satisfying Welch s boun with equality an the other that fins the nearest set of equiangular signatures This algorithm is relate to a metho use by Chu for solving an inverse eigenvalue problem [13] Our algorithm can also be use to fin Grassmannian frames as well as equiangular tight frames We argue that our algorithm converges to a fixe point, an we claim that the class of fixe points contains the esire sequences Detaile proofs of these results are eferre ue to space constraints [14] II SIGNATURE DESIGN PRELIMINARIES Consier the uplink of a single cell, short coe, synchronous CDMA system with N total signatures an a processing gain Let x k enote the 1 signature, coe, or sequence, of user k, normalize as x k =1for k =1,,N We assume that the maximum number of active users allowe in the system is N >1 If the signatures x k form an orthogonal set, the length etermines the allowable number of users It has been shown that nonorthogonal signature sets where N>musers may be necessary to achieve the full sum-capacity of the synchronous single-cell CDMA channel [1] These sequences are calle Welch boun equality sequences [15] since they satisfy the Welch boun on the total square correlation with equality

2 WBE signature sequences have a number of nice properties as summarize in [15], [16] Perhaps the most interesting property is that, using WBE sequences, the interference is uniform across all users [16] The sum total interference in the system is given by k l =k x k, x l 2 N 2 which for WBE sequences is simply N 2 N Using WBE sequences, the total interference power experience by user k is N I(k) = x k, x l 2 1 = N for k =1, 2,,N l=1 (1) an is the same for every user Thus the SINR performance for any user k is simply [ (σ ) 2 1 ( ) ] 1 1 SINR k = s + (2) N σ 2 v an the performance only epens on N an Unfortunately, interference invariance only occurs when the system is fully loae [2], [15], ie N users are active The reason is that a WBE set for N > users almost always ceases to be a WBE set if any M<sequences are remove from or ae to the set [3] Thus if N < N users are active, the whole signature set will nee to be recompute for (, N) an the signatures reassigne or aitional power control will have to compensate for interference inequality III EQUIANGULAR SIGNATURES FOR CDMA SYSTEMS An interesting subclass of WBE signature sequences, known as Grassmannian signatures, retains the interference invariance property even when a subset of signatures are active [3] Grassmannian signatures are constructe from optimal packings of lines on the Grassmann manifol These signature sequences satisfy two important properties: 1) They are equiangular, ie, x k, x l = c for all k, l with k l (3) for some constant c 0 2) They are maximally space, ie c in (3) is as small as possible The equiangular property means that every signature is equally far from every other signature This is the origin of the interference invariance property For example, if N is the set that inexes the active signatures, then the total interference experience by any user k =1, 2,,N is I(k) = x k, x l 2 = c ( N 1) (4) l N /k which only epens on the carinality of N The maximally space property implies that the signature sequence minimizes the maximum angle between the lines generate by x k an x l Let ρ(n,m) := max k,l,k =l x k, x l enote the maximum correlation Grassmannian signatures achieve the lower boun on the maximum correlation for a line packing given by (see [17] for example) ρ(n,) N (N 1) (5) Further, if equality hols in (5), then the signatures are equiangular, maximally space, an form a WBE signature sequence set [4] In general, it is torturous to fin signatures that satisfy (5) with equality Most of the current research has approache the esign problem with algebraic tools A notable triumph of this type is the construction of Kerock coes over Z 2 an Z 4 ue to Calerbank et al [5] Other explicit constructions are iscusse in the articles [4], [3] In the numerical realm, Sloane has use his Gosset software to prouce an stuy sphere packings in real Grassmannian spaces [6] Sloane s algorithms have been extene to complex Grassmannian spaces in [18] We are not aware of any other numerical methos Some examples of signatures that achieve the boun in (5) are available in [4] but generally they are har to fin The reason is that while goo line packings have been tabulate for various an N, these packings o not necessarily maintain the equiangular property On the other han, some equiangular signature sets o not achieve the maximally space property, eg, it is possible to fin five equiangular vectors in R 3 but they are not maximally space In both cases, the resulting packing may not satisfy the WBE property enjoye when equality is satisfie an thus may no longer be capacityoptimal When signature sequences are not available that satisfy (5) with equality, it is not possible to simultaneously obtain a signature sequence that is equiangular, maximally space, an satisfies Welch s boun on the total square correlation with equality Since the equiangular property provies interference invariance, it may be of practical interest to sacrifice the maximally space requirement but yet maintain the constraint that the signature sequence forms a WBE sequence to ensure sum-capacity maximization The objective of this paper is to present an algorithm for fining WBEs that are nearly equiangular Let X = [x 1, x 2,,x N ] be the signature matrix constructe from the signature set It can be shown that a necessary an sufficient conition for a signature sequence to satisfy the Welch boun with equality is that the positive singular values of S are ientical A matrix with this property is calle a tight frame Our goal, then, is to construct a signature matrix X with the following properties i The matrix is a tight frame: XX = α I ii Each column has the correct norm: x n =1 iii The columns are equiangular: x k, x m = c for all k m an some c In this paper we present an algorithm that tries to calculate such sequences that we call equiangular tight frames In the sequel, we summarize the metho an its theoretical behavior

3 IV ALTERNATING PROJECTION PRELIMINARIES Our technique is base on an alternating projection between Property (i) an Properties (ii) (iii) The algorithm attempts to compute a nearby matrix (in terms of the Frobenius norm) that satisfies Properties (i) (iii) Since the Gram matrix X X isplays all of the inner proucts, it is more natural to construct the Gram matrix of an equiangular tight frame than to construct the signature matrix irectly Therefore, our algorithm will alternate between the collection of Hermitian matrices that have the correct spectrum an the collection of Hermitian matrices that have sufficiently small off-iagonal entries Define a collection that contains the Gram matrices of all Nα-tight frames: ef G α = {G C N N : G = G an G has eigenvalues (α,,α, 0,,0)} (6) }{{} The set G α is essentially the Grassmannian manifol that consists of -imensional subspaces of C N [17] One may also ientify the matrices in G α as rank- orthogonal projectors, scale by α Theorem 1 shows how to fin a matrix in G α nearest to an arbitrary Hermitian matrix Theorem 1: Suppose that Z is an N N Hermitian matrix with a unitary factorization UΛU, where the entries of Λ are arrange in algebraically non-increasing orer Let U be the N matrix forme from the first columns of U Then α U U is a matrix in G α that is closest to Z with respect to the Frobenius norm This closest matrix is unique if an only if λ strictly excees λ +1 Proof: See [14] for etails Let H be a close collection of N N Hermitian matrices that satisfy the structural constraint set motivate by the equiangular property: ef H µ = {H C N N : H = H, iag H = m1 an max h mn µ} m =n It may seem more natural to require that the off-iagonal entries have moulus exactly equal to µ, but our experience inicates that the present formulation works better, perhaps because H µ is convex The following proposition shows how to prouce the nearest matrix in H µ Proposition 2: Let Z be an arbitrary matrix With respect to Frobenius norm, the unique matrix in H µ closest to Z has a unit iagonal an off-iagonal entries that satisfy { zmn if z h mn = mn µ an µ e iargzmn otherwise We use i to enote the imaginary unit Proof: The argument is straightforwar The objective of alternating minimization is to fin a solution to the following question Problem 1: Fin a matrix in G α that is minimally istant from H with respect to a given norm If the two sets intersect, any solution to this problem will lie in the intersection Otherwise, the problem requests a tight frame with unit norm columns whose Gram matrix is most nearly equiangular We o not mention the problem of proucing a matrix in H that is nearest to G α because it is not generally possible to factor a matrix in H to obtain a frame with imensions N V STATEMENT OF THE ALGORITHM Practically, implementing to propose alternating minimizing involves alternately enforcing the two aforementione constraint sets until reaching a suitable stopping criterion Convergence is an issue since the tight frame constraint set is non-convex Algorithm 1 (Alternating Projection): INPUT: An arbitrary matrix S 0 The number of iterations J OUTPUT: A signature matrix X J PROCEDURE: 1) Let j =1an H = S0 S 0 2) Fin G j, the Gram matrix nearest to H j 1 in Frobenius norm that has Property (i) 3) Fin H j, the nearest Gram matrix to G j in Frobenius norm that has Properties (ii) an (iii) 4) Increment j Repeat Steps 2 4 until j>j 5) Solve for X J by factoring G J using a finite-step metho such as [19] for example VI SUMMARY OF CONVERGENCE RESULTS The machinery of point-to-set maps is require to unerstan the convergence of this algorithm, so we must refer the reaer to [14] for etails In this section we shall summarize the convergence results A Basic Convergence Results It shoul be clear that alternating projection never increases the istance between successive iterates This oes not mean that it will locate a point of minimal istance between the constraint sets It can be shown, however, that it is globally convergent in a weak sense Define the istance between a point M an a set Y via ist(m, Y )= inf Y Y Y M F Theorem 3 (Global Convergence of Algorithm): Let Y an Z be close sets, one of which is boune Suppose that alternating projection generates a sequence of iterates {(Y j, Z j )} This sequence has at least one accumulation point Every accumulation point lies in Y Z

4 Every accumulation point (Y, Z) satisfies Y Z F = lim j Y j Z j F Every accumulation point (Y, Z) satisfies Y Z F =ist(y, Z )=ist(z, Y ) For a proof of Theorem 3, see the Appenix in [14] The convergence of the propose algorithm is geometric at best [20], [21], [22], [23] This is the major shortfall of alternating projection methos Note that the sequence of iterates may have many accumulation points Moreover, the last conition oes not imply that the accumulation point (Y, Z) is a fixe point of the alternating projection So what are the potential accumulation points of a sequence of iterates? Since the algorithm never increases the istance between successive iterates, the set of accumulation points inclues every pair of matrices in Y Z that lie at minimal istance from each other B Convergence Results Besies the general convergence result, Theorem 3, we also obtain a local convergence result Theorem 4: Assume that the alternating projection between G α an H µ generates a sequence of iterates {(G j, H j )}, an suppose that there is an iteration J uring which G J H J F <N/( 2) The sequence of iterates possesses at least one accumulation point, say (G, H) The accumulation point lies in G α H µ The pair (G, H) is a fixe point of the alternating projection In other wors, if we applie the algorithm to G or to H, every iterate woul equal (G, H) The accumulation point satisfies G HF = lim G j H j j F The component sequences are asymptotically regular, ie G j+1 G j F 0 an H j+1 H j F 0 Either the component sequences both converge in norm, G j G F 0 an H j H F 0, or the set of accumulation points forms a continuum Proof: See the Appenix in [14] A Example Construction VII NUMERICAL EXPERIMENTS First, let us illustrate just how significant a ifference there is between vanilla signature matrices an equiangular signature matrices Here is the Gram matrix of a six-vector, unit-norm tight frame for R 3 : N R R 4 C R R 5 R R 6 R R R 7 C C R 8 C 9 C C 10 R 11 C C 12 C 13 C C R TABLE I N C C C C EQUIANGULAR WBE SIGNATURE SETS The notations R an C respectively inicate that alternating projection was able to compute a real, or complex, equiangular tight frame Note that every real, equiangular tight frame is automatically a complex, equiangular tight frame One perio () means that no real, equiangular tight frame exists, an two perios () mean that no equiangular tight frame exists at all Notice that the inner-proucts between vectors are quite isparate, ranging in magnitue between an These inner proucts correspon to acute angles of 762 an 509 In fact, this tight frame is pretty tame; the largest inner proucts in a unit-norm tight frame can be arbitrarily close to one 1 The Gram matrix of a six-vector, equiangular tight frame for R 3 looks quite ifferent: Every pair of vectors meets at an acute angle of 634 The vectors in this frame can be interprete as the iagonals of an icosaheron [17] B Summary of Basic Constructions We have use alternating projection to compute equiangular tight frames, both real an complex, in imensions two through six The algorithm performe poorly when initialize with ranom vectors, which le us to aopt a more sophisticate approach We begin with many ranom vectors an winnow this collection own by repeately removing whatever vector has the largest inner prouct against another vector It is fast an easy to esign starting points in this manner, yet the results are impressive These calculations are summarize in Table I Alternating projection can locate every real, equiangular tight frame signature matrix in imensions two through six; algebraic consierations eliminate all the 1 To see this, consier a tight frame that contains two copies of an orthonormal basis, where one copy is rotate away from the other by an arbitrarily small angle

5 N Minimum Cor Average Cor St Dev Cor Max Cor Max Cor Packing Max Cor Boun TABLE II NEAR-EQUIANGULAR WBE SIGNATURE SETS Summary of the correlation behavior of specific WBE sequences resulting from the propose algorithm The last three lines compare the maximum correlation of the caniate near-equiangular WBE with the maximum correlation of the best line packing foun for (, N) without the tight frame constraint an the lower boun on the maximum correlation (5) remaining values of N [4] In the complex case, the algorithm was able to compute every equiangular tight frame that we know of Unfortunately, no one has yet evelope necessary conitions on the existence of complex, equiangular tight frames asie from the upper boun, N 2, an so we have been unable to rule out the existence of other ensembles C Overloae System Example We have also constructe some WBEs in imensions of = 2 k for k = 2, 3,,6 an an overloa factor of ten percent The results of this construction are illustrate in Table II Constructions (4, 5) an (8, 9) are exact equiangular tight frames (corresponing to the simplex) In the other cases, the WBEs are only nearly equiangular Because of the tight frame constraint, the maximum correlation is somewhat higher than that of the best line packing for those combinations (without the tight frame constraint), an is larger than the lower boun The stanar eviation of the correlation between two signatures provies a measure of equiangularity Lower values inicate the signatures are more equiangular In the propose examples, there is some variability especially for larger imensions This is because N is not much bigger than thus there are fewer egrees of freeom to enforce the equiangular property For =64an N = 128, though, a construction exists that is equiangular an maximally space [3] VIII CONCLUSION We have propose an alternating minimization that is capable of fining optimal CDMA signature sequences that satisfy equiangular sie constraints an iscusse convergence of the algorithm This algorithm can also be use to solve for unconstraine optimal CDMA signature sequences, sequences with peak-to-average power ratio sie constraints [12], an spectrum constraints A major issue with the propose algorithm is that the resulting sequences are generally complex value an this may lea to implementation challenges Incorporating binary or finite alphabet constraints on the signatures is an interesting topic for future research REFERENCES [1] M Rupf an J L Massey, Optimum sequence multisets for synchronous coe-ivision multiple-access channels, IEEE Trans Inform Theory, vol 40, no 4, pp , July 1994 [2] V Kravcenko, H Boche, F Fitzek, an A Wolisz, No nee for signaling: Investigation of capacity an quality of service for multi-coe CDMA systems usign the wbe ++ approach, in Proc 4th Int Workshop on Mob an Wireless Commun Networks, 2002, pp [3] R W Heath Jr, T Strohmer, an A J Paulraj, Grassmanian signatures for CDMA systems, in Proc of IEEE Global Telecommunications Conf, San Francisco, CA, Dec 2003 [4] T Strohmer an R W Heath Jr, Grassmannian frames with applications to coing an communication, Appl Comp Harmonic Anal, vol 14, no 3, pp , May 2003 [5] A R Calerbank, P J Cameron, W M Kantor, an J J Seiel, Z 4 - 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