Grassmannian Signatures for CDMA Systems - Semantic Scholar
Grassmannian Signatures for CDMA Systems Robert W. Heath Jr.
Thomas Strohmer
Arogyaswami J. Paulraj
Dept. of Elect. and Comp. Engr. The University of Texas at Austin Austin, TX 78712-1084 USA Email: [email protected]
Dept. of Mathematics University of California, Davis Davis, CA 95616 USA Email: [email protected]
Information Systems Laboratory Stanford University Stanford, CA 94305 USA Email: [email protected]
Abstract— Codebooks constructed from Welch bound equality (WBE) sequences have been show to be optimal in terms of sum capacity in synchronous CDMA systems. Unfortunately, these codebooks are a function of the number of active signatures and need to be reassigned as the number of active users changes to maintain optimality. To mitigate the problems caused by the loss of the Welch bound equality property, in this paper we propose a special subclass of WBE sequences for which the interference power experienced by each user depends only on the number of active users and the dimensions of the code. In deference to the relationship with Grassmannian line packing, we refer to this as a Grassmannian signature set. We study the interference properties of this set, comment on the sequence design problem, and illustrate improvements over arbitrary WBE sequence sets via simulation.
I. I NTRODUCTION Direct spread CDMA systems use signature sequences to distinguish between the signals of different users. In singlecell synchronous systems, orthogonal sequences have the advantage that they eliminates interference between users. It has been shown that, however, that non-orthogonal signatures are sum-capacity optimal in synchronous single-cell CDMA systems since CDMA is soft-interference limited (that is, each user can tolerate a certain amount of interference) [1], [2]. Optimal sequence sets are known as Welch bound equality (WBE) sets and have been of considerable interest (see [3] and the references therein). Unfortunately, the WBE sequence set is fundamentally a function of the number of active signatures [3], [4]. Thus whenever a user leaves or a new user arrives, the codebook changes and all the codes must be reassigned. Alternatively, the subset of remaining sequences will no longer be optimal. It is of interest to find WBE sequence sets that do not have to be reassigned when users enter or leave the system. In this paper we consider the class of sequences that achieve the maximum Welch bound with equality [3], [5]. Due to the relation with the subspace packing problem, we refer to these sequences as Grassmannian sequence sets. We show that these sequences are equiangular thus the interference experienced by each code is the same and is only a function of the number of active codes and the dimensionality of the codebook and not the code. We find a closed form expression for the interference 1 This material is based in part upon work supported by the Texas Advanced Research (Technology) Program under Grant No. 003658-0614-2001, NSF grant DMS 0208568, and the Samsung Institute of Applied Technology.
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experienced by each code and show this is equivalent to the average interference derived when a subset of WBEs are sequences are active [4]. Using the relationship with line packing we present a practical construction using results from [6]. Through simulations of random subsets of active users, we illustrate the benefits of Grassmannian sequences over arbitrarily chosen WBE sequence sets in terms of variability of interference, condition number of the Gram matrix constructed from the signatures, and proximity to the Welch bound on total squared correlation. Much of the prior work on quasi-orthogonal signature construction for CDMA systems considers signatures that are constructed from sequences with elements from a complex constellation. This simplifies implementation and leads to many interesting constructions [7]–[9]. In this paper, however, we follow the approach of Welch [5] and consider the most general case when signatures have elements that are drawn from either the real or the complex field. Algorithms for finding WBE sequences have appeared in [2], [10], [11]. The optimality of WBE sets when minimum mean squared error (MMSE) decoding is used at the receiver for the uplink and downlink cases and a variety of power control strategies is derived in [12]. This paper is organized as follows. In Section 2 we present some background and motivation for this problem. In Section 3 we propose Grassmannian sequence sets and comment on their interference invariance in Section 4. In Section 5 we comment on the construction of such sequences while in Section 6 we illustrate their utility over arbitrarily chosen WBE sequence sets. Finally, in Section 7 we present some directions for future work.
II. BACKGROUND AND M OTIVATION Consider the downlink of a single cell, short code, synchronous CDMA system with N total signatures and a processing gain m. Let sk denote the m × 1 signature, code, or sequence, of user k and assume that there are N ≥ m > 1 active users in the system. In this paper we consider explicitly the case as in [1] where all the received powers are the same and thus fading can be neglected. Assuming perfect synchronization and chip-rate sampling, during one symbol
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period the receiver for user k observes the m × 1 vector N yk = σ s sn bn (1) + vk n=1
where bn is the complex symbol transmitted by the n−th user, σs2 is the transmitted signal energy, h is the complex channel gain, and vk is an m × 1 vector sample of an independent identically distributed (i.i.d.) circularly symmetric complex Gaussian random vector with distribution CN (0, σv2 I). Assume the normalization sk = 1 for k = 1, . . . , N . If the signatures sk form an orthogonal set, the length m determines the allowable number of users. It has been shown that nonorthogonal signature sets allowing N > m users may be necessary to achieve the full sum-capacity of the synchronous single-cell CDMA channel [1]. These sequences are the so-called Welch bound equality sequences [3] since they satisfy the Welch bound on the total squared correlation with equality A(S) =
N N k=1 l=1
|sk , sl |2 ≥
N2 . m
(2)
From [3], [13] WBE sequences have a number of nice properties. For example, they have been shown to maximize the sum-capacity of the synchronous CDMA channel [1]. Further, for WBE sequences the MMSE receiver coincides with the matched filter receiver [2]. Perhaps the most interesting property is that, using WBE sequences, the interference is uniform across all users [13]. The sum total interference in the system is given by A(S)−N 2 2 which for WBE sequences is simply Nm − N . Using WBE sequences, the total interference power experienced by user k is N
N −m for k = 1, 2, . . . , N m l=1 (3) and is the same for every user. Thus the SINR performance for any user k is simply −1 −1 −1 σs2 m + (4) SINRk = σv2 N −m
I(k) =
|sk , sl |2 − 1 =
and the performance only depends on N and m. Despite the interesting mathematical implications in synchronous CDMA systems, as mentioned in [3] and studied in more detail in [4], [14], WBE sequences seem less useful in practical implementations. Even if one constructs WBE that satisfy additional properties they have a serious drawback. If we design a WBE set for N > m users, then this set almost always ceases to be a WBE set if any M < m sequences are removed from or added to the set as shown in the following. Theorem 1: Let S = [s1 , . . . , sN ] be a set of WBE sequences of length m and assume N > m. If we remove any M < m sequences from or add any M < m unit-norm sequences to this set, then the resulting set has no longer the WBE property.
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Proof: See [15] for details. 2 This means when the number of users changes in a cell, in many cases a WBE set designed for N active users is no longer optimal. Essentially, if the matched filter is used with a subset of the WBE sequences active then the uniformity of the SIR is typically lost. This leads to the undesirable property that users would see different amounts of interference as a function of their sequence assignment which can result in capacity or bit error probability degradations [4]. Thus a system that fully exploits WBE sequences would need (i) a set of sequences for every possible N and (ii) would need to reassign all sequences every time a user arrived or departed from the system. To mitigate the problems caused by the loss of the WBE property, in this paper we study a special subclass of WBE ¯ (the sequences for which the SINR only depends on N number of active users) and m thus retaining the interference invariance for subsets of active users. III. G RASSMANNIAN S IGNATURES FOR CDMA S YSTEMS The importance of general WBE sequence sets for S-CDMA has overshadowed the utility of a subclass of these sequences known as maximum WBE (MWBE) sequences. The Welch bound as originally stated is a lower bound on the maximum correlation for a set of N signature sequences sk in Cm [5] defined as (5) ρ(N, m) := max |sk , sl |. k,l,k=l
This problem is equivalent to finding a lower bound on the maximum angle between the lines generated by sk and sl . In particular, this problem relates to the notion of packings of subspaces. Recall the following definition. Definition 2: The Grassmannian space G(m, n) is the set of all n-dimensional subspaces of the space E m , where E = R or E = C. Often the Grassmannian space is defined for E = R only. The Grassmannian packing problem is the problem of finding the best packing of N n-dimensional subspaces in E m (see e.g. [16], [17]). In other words, it is the problem of finding N points in G(m, n) so that the minimal distance between any two of them is as large as possible. In the general case one has to define the angle between two subspaces, however, in this paper we consider n = 1 exclusively which reduces to the angle between two lines. Thus the subspaces are lines through the origin in E m and the goal is to arrange the lines such that the angle between any two of the lines becomes as large as possible. Given these definitions, the following theorem summarizes the findings of researchers working in a variety of related fields. Theorem 3: Let S = {s1 , . . . , sN } be a set of unit vectors in E m with N ≥ m. Then N −m . (6) ρ(N, m) ≥ m(N − 1) If E = R then equality in (6) can only hold if N ≤ m(m + 1)/2 and if E = C then equality in (6) can only hold if
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N ≤ m2 . Furthermore if equality holds in (6) then the vectors in S are equiangular meaning that |sk , sl | = c for any k = l for some positive c. Proof: See [5], [16], [18] for example. 2 According to [16] the bound (6) is essentially Rankin’s bound on spherical codes [19], also known as the Simplex bound. It has been rediscovered (sometimes in a somewhat different form) by researchers in areas ranging from coding theory [5], to graph theory [18] and algebraic geometry [20]. More details on the connections between these applications, and their relation to a problem in frame theory, is available in [6]. It is readily checked that the bound in (6) is simply Welch’s bound on the maximum correlation [5]. Signature sets that satisfy (6) with equality are thus optimal packings in the sense that they achieve the largest possible uniform packing. In deference to the relation with subspace packing, we use the term Grassmannian sequences though the term MWBE sequence sets from [3] is equivalent. It is clear, that Grassmannian sequences are a subclass of WBE sequences since N N
|sk , sl |2
= N+
k=1 l=1
N N
|sk , sl |2
k=1 l=1,l=k
= N +N
N −1 m(N − 1)
N2 (7) m and thus have all the properties of WBE sequences in terms of capacity and uniformity of the SIR mentioned in Section III. =
IV. I NTERFERENCE I NVARIANCE OF G RASSMANNIAN S EQUENCES Perhaps the most interesting property of Grassmannian sequence sets is that the total interference power for every sequence is only a function of the current number of active sequence and the original dimensionality of the codebook. This is a byproduct of the equiangular property and is stated in the following theorem. Theorem 4: The total interference power for any equiangular sequence set is identical for all sequences and depends only on the total number of active sequences. Proof: Let N denote the set that indexes the active sequences. For an arbitrary active sequence k, the interference power is |sk , sn |2 = (|N | − 1)c2 (8) I(k) = n∈N ,n=k
where c is the equiangular constant, i.e., c = |sk , sl | for any k = l and the result follows from the definition of equiangular. Note that (8) is independent of k and depends only on the number of active users given by the cardinality of N . 2 In fact, there are many possible sets of equiangular sequences. A result of this theorem, which is a byproduct of Theorem 3, is that Grassmannian sequences are the best of
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all equiangular sequences since they achieve the lowest bound on the maximum correlation with equality (and thus have the smallest possible c). Thus for Grassmannian sequence sets, the interference power experienced by the k th user is (|N | − 1) (N − m) (9) I(k) = |sk , sl |2 = m (N − 1) l∈N ,l=k
which is the same for k = 1, 2, . . . , N . Further, the SINR for user k (or any user) is simply written as −1 −1 −1 σs2 m(N − 1) + SINRk = σv2 (N − m)(|N | − 1) (10) and the performance again only depends on the number of active sequences |N | and the original dimensionality of the code given by N and m. V. C ONSTRUCTION OF G RASSMANNIAN S EQUENCES The relationship between Grassmannian signature sequences and the classical subspace-packing problem reveals the challenges in finding good signature sets, i.e. it hints that finding optimal signature sets may be difficult. The connection, however, allows us to take advantage of the fruits of prior researchers that have cataloged tables with the best packings to date (see [16], or [21] for details). Unfortunately, most of these packings are in the real space and for relatively small dimensions [21]. Therefore the problem of finding complex constructions is still largely open. In some special cases we can find constructions for certain sets of parameters. Here we summarize one useful construction from [6] that uses conference matrices and skew-symmetric conference matrices. Lemma 5: Let C2m be √ a (skew)symmetric conference matrix and denote α := 1/ 2m − 1. Compute αC2m + I2m if E = R, . (11) R= R = jαC2m + I2m if E = C. Let w1 , . . . , w2m be the eigenvectors of R. Then the 2m vectors v1 , . . . , v2m in E m given by √ k = 1, . . . , 2m, (12) vk := 2[w1 (k), . . . , wm (k)]T , constitute a set of vectors that achieve the bound (6). Proof: See [6] for details. 2 A necessary condition for the existence of a symmetric conference matrix Cn is that n = 2 mod 4. A sufficient condition due to Paley [22] is n = pα + 1 where p is a prime number and α ∈ N. In this case it is not difficult to explicitly construct the conference matrix. A sufficient condition for the existence of a skew-symmetric conference matrix Cn is that n = 2k for k ∈ N. Here is a simple recursive procedure to construct a skew-symmetric conference matrix whose size is a power of two. Initialize
0 −1 , (13) C2 = 1 0
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and compute recursively
Cm C2m = Cm + Im
45
Cm − Im , −Cm
WBE MWBE
40
(14) 35
then C2m is a skew-symmetric conference matrix. This construction is reminiscent of the construction of Hadamard matrices. Indeed, a simple calculation shows that Cm + Im is a skew-symmetric Hadamard matrix. Hence these explicit constructions of conference matrices in combination with Lemma 5 provide a convenient way to construct a set of 2m vectors in E m with minimal correlation.
Average SIR (dB)
30
25
20
15
10
5
VI. C OMPARISON WITH A RBITRARY WBE S EQUENCE S ETS To understand the value of the equiangular property of Grassmannian sequence sets, consider the following practical scenario. Suppose that the codebook is designed for a total ¯ = |N | ≤ N codes are in use. Since of N codes but N Grassmannian sequence sets are also WBE sequence sets, ¯ = N it follows from (3) that the interference when N power is constant and is the same for each sequence. When ¯ < N , however, a non-Grassmannian sequence set m ≤ N does not generally retain this property (see the example that follows) since satisfying the Welch bound on sum squared correlation with equality in (2) is not a sufficient condition for a sequence set to be equiangular. To illustrate the difference between Grassmannian sequence sets and a general WBE set, we consider the following example. Suppose that N = 256 and m = 128 using either a randomly chosen WBE signature set or a Grassmannian signature set derived from the construction in Lemma 5. In Fig. 1 we plot the SIR as a function of the number of active ¯ = 2 to N ¯ = 256. For the WBE case we plot the codes for N max, mean, and min SIR over the number of active codes. To remove the influence of code selection, given the first signature ¯ > 1 we computed the SIR for is always active, for each N ¯ − 1 randomly chosen signatures. We repeated this a set of N 20 times to average the max, min, and mean SIR. For the Grassmannian case the average SIR is equal to the maximum SIR and the minimum SIR, irrespectively of the signatures added, due to the equiangular property. As illustrated in Fig. 1, the effect of using an arbitrary WBE sequence set is to increase the variance of the interference. Since it is not practical to predict the activity of all the users, this variation will need to be compensated by additional power control. From Fig. 1, the average SIRs for both signature sets are similar throughout the range of added signatures. While at first this seems surprising, this result can actually be deduced from [4, eq. (9)] (changing to our notation) in which this expectation for WBE sequence sets with a fraction of users active is explicitly computed. In fact, given this relationship, we can study the bit error rate degradation using [4, eq. (10)] except that this is an even better approximation in our case since we do not average over all possible sets of active users.
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0
50
100 150 number of additional vectors
200
250
Fig. 1. Comparison of SIR as the number of active codes increases. For the Grassmannian case the SIR is the same for all signatures. For the WBE case, for each point, we plot the minimum, maximum, and average SIR over the group of active signatures.
Another way to understand the benefits of Grassmannian sequences is to examine the condition number of the resulting sequence matrix S, the matrix whose columns consist of the active signatures, when more and more active codes are added. This provides a measure of “distance” of SS∗ (the Gram matrix of S to an identity matrix). In essence it indicates how close the set is to being a WBE signature set since SS∗ = N m Im is a necessity for a WBE set with N users and dimension m [13]. Following [23, eq. (5.8.10)], it also gives some measure of separability of the signatures when using linear receivers such as the MMSE receiver (see also the discussion in [6]). ¯ ≥ m. We begin To make this comparison, we consider N with a baseline set of m sequences that are optimally selected such that the condition number of their Gram matrix is as small as possible. The idea behind this is that the system may still mainly designed for m users, thus it makes sense to select in advance codes for the most frequently used case. Then we successively add to each set a randomly chosen sequence from the remaining sequences until we reach the full 2mset of codes. In each step we compute the condition number of the sequence matrix and the squared Frobenius norm of the corresponding Gram matrix (i.e., the sum-correlation) for both sets. The latter is compared to the optimal value given by the Welch bound. The results are shown in Fig. 2-Fig. 3. When comparing the condition numbers in Fig. 2 we see that Grassmannian sequences clearly outperform WBE sequences and are relatively insensitive to the number of active codes. If a WBE set were designed for each subset ¯ | then the condition number would always be one. size |N The fact that Grassmannian sequences are not as good as redesigning a WBE for each possible number of users is offset by their flexibility in not having to continually reassign codes. Examining Fig. 3 shows that both sets have a similar distance to the ideal WBE set custom designed for the number of active signatures. Thus from the point-of-view of total correlation,
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interest to consider the equivalent of generalized WBE sequences [2] where it is expected that Grassmannian sequences will have a similar roll but only for the non-oversized users.
35 WBE MWBE 30
R EFERENCES
condition number
25
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10
5
20
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60 80 number of additional vectors
100
120
Fig. 2. Comparison of condition number of sequence matrices, when more and more users are successively added to the system.
500 WBE MWBE bound 450
total sum−squared correlation
400
350
300
250
200
150 20
40
60 80 number of additional vectors
100
120
Fig. 3. Comparison of the sum correlation of both sets, when more and more users are successively added to the system.
enforcing the maximally spaced equal angular property is no better or worse than an arbitrary WBE set. VII. C ONCLUSION AND F UTURE W ORK In this paper we studied an important subclass of WBE sequences known as Grassmannian sequences. We showed that this class of signatures has a pleasing interference invariance property for subsets of active users. While we considered sequences with arbitrary entries, it is of interest to study sequences from finite alphabets. It is unclear when equiangular sequence sets might exist in this case. Further, in this paper we only considered subsets of WBE sequences. It is also of
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[1] M. Rupf and J. L. Massey, “Optimum sequence multisets for synchronous code-division multiple-access channels,” IEEE Trans. Info. Theory, vol. 40, no. 4, pp. 1261–1266, 1994. [2] P. Viswanath and V. Anantharam, “Optimal sequences and sum capacity of synchronous CDMA systems,” IEEE Trans. Info. Theory, vol. 45, no. 6, pp. 1984–1991, Sept. 1999. [3] D. V. Sarwate, “Meeting the Welch bound with equality,” in Sequences and their applications (Singapore, 1998). London: Springer, 1999, pp. 79–102. [4] V. Kravcenko, H. Boche, F. Fitzek, and A. Wolisz, “No need for signaling: Investigation of capacity and quality of service for multi-code cdma systems usign the wbe ++ approach,” in Proc. 4th Int. Workshop on Mob. and Wireless Commun. Networks, 2002, pp. 110–114. [5] L. Welch, “Lower bounds on the maximum cross-correlation of signals,” IEEE Trans. Info. Theory, vol. 20, pp. 397–399, 1974. [6] T. Strohmer and R. W. Heath Jr., “Grassmannian frames with applications to coding and communications,” Applied and Computational Harmonic Analysis, vol. 14, no. 3, pp. 257–275, July 2003. [7] A. Shanbhag and J. Holtzman, “Optimal QPSK modulated quasiorthogonal functions for IS-2000,” in IEEE 6-th Int. Symp. Spread Spectrum Tech. & Appli., New Jersey, USA, Sept. 2000, pp. 756–760. [8] H. D. Schotten and H. Hadinejad-Mahram, “Analysis of a CDMA downlink with non-orthogonal spreading sequences for fading channels,” in Proc. of Veh. Tech. Conf., 2000, pp. 1782–1786. [9] K. Yang, Y.-K. Kim, and P. V. Kumar, “Quasi-orthogonal sequences for code-division multiple-access systems,” IEEE Trans. Info. Theory, vol. 46, no. 3, pp. 982–993, 2000. [10] S. Ulukus and R. D. Yates, “Iterative construction of optimum signature sequence sets in synchronous CDMA systems,” IEEE Trans. Info. Theory, vol. 47, no. 5, pp. 1989–1998, 2001. [11] J. A. Tropp, R. W. Heath Jr., and T. Strohmer, “An alternating projection algorithm for constructing quasi-orthogonal CDMA signature sequences,” in Proc. Int. Symp. on Info. Theo., 2003. [12] P. Viswanath, V. Anantharam, and D. N. C. Tse, “Optimal sequences, power control, and user capacity of synchronous CDMA systems with linear MMSE multiuser receivers,” IEEE Trans. Info. Theory, vol. 45, no. 6, pp. 1968–1983, 1999. [13] J. L. Massey and T. Mittelholzer, “Welch’s bound and sequence sets for code division multiple-access systems,” in Sequences II: Methods in Communication, Security, and Computer Science. New York: SpringerVerlag, 1993, pp. 63–78. [14] T. Strohmer, R. W. Heath Jr., and A. J. Paulraj, “On the design of optimal sequences for CDMA systems,” in Proceedings of the Thirty-Sixth Asil. Conf. on Sig., Sys. and Comp., vol. 2, Nov. 2002, pp. 1434–1438. [15] R. W. Heath Jr., T. Strohmer, and A. J. Paulraj, “On quasi-orthogonal signatures for CDMA systems,” submitted to IEEE Trans. Inform. Theory September 2002. [16] J. H. Conway, R. H. Hardin, and N. J. A. Sloane, “Packing lines, planes, etc.: packings in Grassmannian spaces,” Experiment. Math., vol. 5, no. 2, pp. 139–159, 1996. [17] A. R. Calderbank, R. H. Hardin, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “A group-theoretic framework for the construction of packings in Grassmannian spaces,” J. Algebraic Combin., vol. 9, no. 2, pp. 129– 140, 1999. [18] P. Delsarte, J. M. Goethals, and J. J. Seidel, “Bounds for systems of lines and Jacobi poynomials,” Philips Res. Repts, vol. 30, no. 3, pp. 91∗ –105∗ , 1975, issue in honour of C.J. Bouwkamp. [19] R. Rankin, “The closest packing of spherical caps in n dimensions,” Proc. Glasgow Math. Assoc., vol. 2, pp. 139–144, 1955. [20] M. Rosenfeld, “In praise of the Gram matrix,” in The mathematics of Paul Erd˝os, II. Berlin: Springer, 1997, pp. 318–323. [21] N. J. A. Sloane, “Packings in Grassmannian spaces,” http://www.research.att.com/ ˜njas/grass/index.html. [22] R. Paley, “On orthogonal matrices,” J. Math. Phys., vol. 12, pp. 311–320, 1933. [23] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge University Press, 1985.
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Thomas Strohmer
Arogyaswami J. Paulraj
Dept. of Elect. and Comp. Engr. The University of Texas at Austin Austin, TX 78712-1084 USA Email: [email protected]
Dept. of Mathematics University of California, Davis Davis, CA 95616 USA Email: [email protected]
Information Systems Laboratory Stanford University Stanford, CA 94305 USA Email: [email protected]
Abstract— Codebooks constructed from Welch bound equality (WBE) sequences have been show to be optimal in terms of sum capacity in synchronous CDMA systems. Unfortunately, these codebooks are a function of the number of active signatures and need to be reassigned as the number of active users changes to maintain optimality. To mitigate the problems caused by the loss of the Welch bound equality property, in this paper we propose a special subclass of WBE sequences for which the interference power experienced by each user depends only on the number of active users and the dimensions of the code. In deference to the relationship with Grassmannian line packing, we refer to this as a Grassmannian signature set. We study the interference properties of this set, comment on the sequence design problem, and illustrate improvements over arbitrary WBE sequence sets via simulation.
I. I NTRODUCTION Direct spread CDMA systems use signature sequences to distinguish between the signals of different users. In singlecell synchronous systems, orthogonal sequences have the advantage that they eliminates interference between users. It has been shown that, however, that non-orthogonal signatures are sum-capacity optimal in synchronous single-cell CDMA systems since CDMA is soft-interference limited (that is, each user can tolerate a certain amount of interference) [1], [2]. Optimal sequence sets are known as Welch bound equality (WBE) sets and have been of considerable interest (see [3] and the references therein). Unfortunately, the WBE sequence set is fundamentally a function of the number of active signatures [3], [4]. Thus whenever a user leaves or a new user arrives, the codebook changes and all the codes must be reassigned. Alternatively, the subset of remaining sequences will no longer be optimal. It is of interest to find WBE sequence sets that do not have to be reassigned when users enter or leave the system. In this paper we consider the class of sequences that achieve the maximum Welch bound with equality [3], [5]. Due to the relation with the subspace packing problem, we refer to these sequences as Grassmannian sequence sets. We show that these sequences are equiangular thus the interference experienced by each code is the same and is only a function of the number of active codes and the dimensionality of the codebook and not the code. We find a closed form expression for the interference 1 This material is based in part upon work supported by the Texas Advanced Research (Technology) Program under Grant No. 003658-0614-2001, NSF grant DMS 0208568, and the Samsung Institute of Applied Technology.
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experienced by each code and show this is equivalent to the average interference derived when a subset of WBEs are sequences are active [4]. Using the relationship with line packing we present a practical construction using results from [6]. Through simulations of random subsets of active users, we illustrate the benefits of Grassmannian sequences over arbitrarily chosen WBE sequence sets in terms of variability of interference, condition number of the Gram matrix constructed from the signatures, and proximity to the Welch bound on total squared correlation. Much of the prior work on quasi-orthogonal signature construction for CDMA systems considers signatures that are constructed from sequences with elements from a complex constellation. This simplifies implementation and leads to many interesting constructions [7]–[9]. In this paper, however, we follow the approach of Welch [5] and consider the most general case when signatures have elements that are drawn from either the real or the complex field. Algorithms for finding WBE sequences have appeared in [2], [10], [11]. The optimality of WBE sets when minimum mean squared error (MMSE) decoding is used at the receiver for the uplink and downlink cases and a variety of power control strategies is derived in [12]. This paper is organized as follows. In Section 2 we present some background and motivation for this problem. In Section 3 we propose Grassmannian sequence sets and comment on their interference invariance in Section 4. In Section 5 we comment on the construction of such sequences while in Section 6 we illustrate their utility over arbitrarily chosen WBE sequence sets. Finally, in Section 7 we present some directions for future work.
II. BACKGROUND AND M OTIVATION Consider the downlink of a single cell, short code, synchronous CDMA system with N total signatures and a processing gain m. Let sk denote the m × 1 signature, code, or sequence, of user k and assume that there are N ≥ m > 1 active users in the system. In this paper we consider explicitly the case as in [1] where all the received powers are the same and thus fading can be neglected. Assuming perfect synchronization and chip-rate sampling, during one symbol
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period the receiver for user k observes the m × 1 vector N yk = σ s sn bn (1) + vk n=1
where bn is the complex symbol transmitted by the n−th user, σs2 is the transmitted signal energy, h is the complex channel gain, and vk is an m × 1 vector sample of an independent identically distributed (i.i.d.) circularly symmetric complex Gaussian random vector with distribution CN (0, σv2 I). Assume the normalization sk = 1 for k = 1, . . . , N . If the signatures sk form an orthogonal set, the length m determines the allowable number of users. It has been shown that nonorthogonal signature sets allowing N > m users may be necessary to achieve the full sum-capacity of the synchronous single-cell CDMA channel [1]. These sequences are the so-called Welch bound equality sequences [3] since they satisfy the Welch bound on the total squared correlation with equality A(S) =
N N k=1 l=1
|sk , sl |2 ≥
N2 . m
(2)
From [3], [13] WBE sequences have a number of nice properties. For example, they have been shown to maximize the sum-capacity of the synchronous CDMA channel [1]. Further, for WBE sequences the MMSE receiver coincides with the matched filter receiver [2]. Perhaps the most interesting property is that, using WBE sequences, the interference is uniform across all users [13]. The sum total interference in the system is given by A(S)−N 2 2 which for WBE sequences is simply Nm − N . Using WBE sequences, the total interference power experienced by user k is N
N −m for k = 1, 2, . . . , N m l=1 (3) and is the same for every user. Thus the SINR performance for any user k is simply −1 −1 −1 σs2 m + (4) SINRk = σv2 N −m
I(k) =
|sk , sl |2 − 1 =
and the performance only depends on N and m. Despite the interesting mathematical implications in synchronous CDMA systems, as mentioned in [3] and studied in more detail in [4], [14], WBE sequences seem less useful in practical implementations. Even if one constructs WBE that satisfy additional properties they have a serious drawback. If we design a WBE set for N > m users, then this set almost always ceases to be a WBE set if any M < m sequences are removed from or added to the set as shown in the following. Theorem 1: Let S = [s1 , . . . , sN ] be a set of WBE sequences of length m and assume N > m. If we remove any M < m sequences from or add any M < m unit-norm sequences to this set, then the resulting set has no longer the WBE property.
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Proof: See [15] for details. 2 This means when the number of users changes in a cell, in many cases a WBE set designed for N active users is no longer optimal. Essentially, if the matched filter is used with a subset of the WBE sequences active then the uniformity of the SIR is typically lost. This leads to the undesirable property that users would see different amounts of interference as a function of their sequence assignment which can result in capacity or bit error probability degradations [4]. Thus a system that fully exploits WBE sequences would need (i) a set of sequences for every possible N and (ii) would need to reassign all sequences every time a user arrived or departed from the system. To mitigate the problems caused by the loss of the WBE property, in this paper we study a special subclass of WBE ¯ (the sequences for which the SINR only depends on N number of active users) and m thus retaining the interference invariance for subsets of active users. III. G RASSMANNIAN S IGNATURES FOR CDMA S YSTEMS The importance of general WBE sequence sets for S-CDMA has overshadowed the utility of a subclass of these sequences known as maximum WBE (MWBE) sequences. The Welch bound as originally stated is a lower bound on the maximum correlation for a set of N signature sequences sk in Cm [5] defined as (5) ρ(N, m) := max |sk , sl |. k,l,k=l
This problem is equivalent to finding a lower bound on the maximum angle between the lines generated by sk and sl . In particular, this problem relates to the notion of packings of subspaces. Recall the following definition. Definition 2: The Grassmannian space G(m, n) is the set of all n-dimensional subspaces of the space E m , where E = R or E = C. Often the Grassmannian space is defined for E = R only. The Grassmannian packing problem is the problem of finding the best packing of N n-dimensional subspaces in E m (see e.g. [16], [17]). In other words, it is the problem of finding N points in G(m, n) so that the minimal distance between any two of them is as large as possible. In the general case one has to define the angle between two subspaces, however, in this paper we consider n = 1 exclusively which reduces to the angle between two lines. Thus the subspaces are lines through the origin in E m and the goal is to arrange the lines such that the angle between any two of the lines becomes as large as possible. Given these definitions, the following theorem summarizes the findings of researchers working in a variety of related fields. Theorem 3: Let S = {s1 , . . . , sN } be a set of unit vectors in E m with N ≥ m. Then N −m . (6) ρ(N, m) ≥ m(N − 1) If E = R then equality in (6) can only hold if N ≤ m(m + 1)/2 and if E = C then equality in (6) can only hold if
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N ≤ m2 . Furthermore if equality holds in (6) then the vectors in S are equiangular meaning that |sk , sl | = c for any k = l for some positive c. Proof: See [5], [16], [18] for example. 2 According to [16] the bound (6) is essentially Rankin’s bound on spherical codes [19], also known as the Simplex bound. It has been rediscovered (sometimes in a somewhat different form) by researchers in areas ranging from coding theory [5], to graph theory [18] and algebraic geometry [20]. More details on the connections between these applications, and their relation to a problem in frame theory, is available in [6]. It is readily checked that the bound in (6) is simply Welch’s bound on the maximum correlation [5]. Signature sets that satisfy (6) with equality are thus optimal packings in the sense that they achieve the largest possible uniform packing. In deference to the relation with subspace packing, we use the term Grassmannian sequences though the term MWBE sequence sets from [3] is equivalent. It is clear, that Grassmannian sequences are a subclass of WBE sequences since N N
|sk , sl |2
= N+
k=1 l=1
N N
|sk , sl |2
k=1 l=1,l=k
= N +N
N −1 m(N − 1)
N2 (7) m and thus have all the properties of WBE sequences in terms of capacity and uniformity of the SIR mentioned in Section III. =
IV. I NTERFERENCE I NVARIANCE OF G RASSMANNIAN S EQUENCES Perhaps the most interesting property of Grassmannian sequence sets is that the total interference power for every sequence is only a function of the current number of active sequence and the original dimensionality of the codebook. This is a byproduct of the equiangular property and is stated in the following theorem. Theorem 4: The total interference power for any equiangular sequence set is identical for all sequences and depends only on the total number of active sequences. Proof: Let N denote the set that indexes the active sequences. For an arbitrary active sequence k, the interference power is |sk , sn |2 = (|N | − 1)c2 (8) I(k) = n∈N ,n=k
where c is the equiangular constant, i.e., c = |sk , sl | for any k = l and the result follows from the definition of equiangular. Note that (8) is independent of k and depends only on the number of active users given by the cardinality of N . 2 In fact, there are many possible sets of equiangular sequences. A result of this theorem, which is a byproduct of Theorem 3, is that Grassmannian sequences are the best of
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all equiangular sequences since they achieve the lowest bound on the maximum correlation with equality (and thus have the smallest possible c). Thus for Grassmannian sequence sets, the interference power experienced by the k th user is (|N | − 1) (N − m) (9) I(k) = |sk , sl |2 = m (N − 1) l∈N ,l=k
which is the same for k = 1, 2, . . . , N . Further, the SINR for user k (or any user) is simply written as −1 −1 −1 σs2 m(N − 1) + SINRk = σv2 (N − m)(|N | − 1) (10) and the performance again only depends on the number of active sequences |N | and the original dimensionality of the code given by N and m. V. C ONSTRUCTION OF G RASSMANNIAN S EQUENCES The relationship between Grassmannian signature sequences and the classical subspace-packing problem reveals the challenges in finding good signature sets, i.e. it hints that finding optimal signature sets may be difficult. The connection, however, allows us to take advantage of the fruits of prior researchers that have cataloged tables with the best packings to date (see [16], or [21] for details). Unfortunately, most of these packings are in the real space and for relatively small dimensions [21]. Therefore the problem of finding complex constructions is still largely open. In some special cases we can find constructions for certain sets of parameters. Here we summarize one useful construction from [6] that uses conference matrices and skew-symmetric conference matrices. Lemma 5: Let C2m be √ a (skew)symmetric conference matrix and denote α := 1/ 2m − 1. Compute αC2m + I2m if E = R, . (11) R= R = jαC2m + I2m if E = C. Let w1 , . . . , w2m be the eigenvectors of R. Then the 2m vectors v1 , . . . , v2m in E m given by √ k = 1, . . . , 2m, (12) vk := 2[w1 (k), . . . , wm (k)]T , constitute a set of vectors that achieve the bound (6). Proof: See [6] for details. 2 A necessary condition for the existence of a symmetric conference matrix Cn is that n = 2 mod 4. A sufficient condition due to Paley [22] is n = pα + 1 where p is a prime number and α ∈ N. In this case it is not difficult to explicitly construct the conference matrix. A sufficient condition for the existence of a skew-symmetric conference matrix Cn is that n = 2k for k ∈ N. Here is a simple recursive procedure to construct a skew-symmetric conference matrix whose size is a power of two. Initialize
0 −1 , (13) C2 = 1 0
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and compute recursively
Cm C2m = Cm + Im
45
Cm − Im , −Cm
WBE MWBE
40
(14) 35
then C2m is a skew-symmetric conference matrix. This construction is reminiscent of the construction of Hadamard matrices. Indeed, a simple calculation shows that Cm + Im is a skew-symmetric Hadamard matrix. Hence these explicit constructions of conference matrices in combination with Lemma 5 provide a convenient way to construct a set of 2m vectors in E m with minimal correlation.
Average SIR (dB)
30
25
20
15
10
5
VI. C OMPARISON WITH A RBITRARY WBE S EQUENCE S ETS To understand the value of the equiangular property of Grassmannian sequence sets, consider the following practical scenario. Suppose that the codebook is designed for a total ¯ = |N | ≤ N codes are in use. Since of N codes but N Grassmannian sequence sets are also WBE sequence sets, ¯ = N it follows from (3) that the interference when N power is constant and is the same for each sequence. When ¯ < N , however, a non-Grassmannian sequence set m ≤ N does not generally retain this property (see the example that follows) since satisfying the Welch bound on sum squared correlation with equality in (2) is not a sufficient condition for a sequence set to be equiangular. To illustrate the difference between Grassmannian sequence sets and a general WBE set, we consider the following example. Suppose that N = 256 and m = 128 using either a randomly chosen WBE signature set or a Grassmannian signature set derived from the construction in Lemma 5. In Fig. 1 we plot the SIR as a function of the number of active ¯ = 2 to N ¯ = 256. For the WBE case we plot the codes for N max, mean, and min SIR over the number of active codes. To remove the influence of code selection, given the first signature ¯ > 1 we computed the SIR for is always active, for each N ¯ − 1 randomly chosen signatures. We repeated this a set of N 20 times to average the max, min, and mean SIR. For the Grassmannian case the average SIR is equal to the maximum SIR and the minimum SIR, irrespectively of the signatures added, due to the equiangular property. As illustrated in Fig. 1, the effect of using an arbitrary WBE sequence set is to increase the variance of the interference. Since it is not practical to predict the activity of all the users, this variation will need to be compensated by additional power control. From Fig. 1, the average SIRs for both signature sets are similar throughout the range of added signatures. While at first this seems surprising, this result can actually be deduced from [4, eq. (9)] (changing to our notation) in which this expectation for WBE sequence sets with a fraction of users active is explicitly computed. In fact, given this relationship, we can study the bit error rate degradation using [4, eq. (10)] except that this is an even better approximation in our case since we do not average over all possible sets of active users.
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0
50
100 150 number of additional vectors
200
250
Fig. 1. Comparison of SIR as the number of active codes increases. For the Grassmannian case the SIR is the same for all signatures. For the WBE case, for each point, we plot the minimum, maximum, and average SIR over the group of active signatures.
Another way to understand the benefits of Grassmannian sequences is to examine the condition number of the resulting sequence matrix S, the matrix whose columns consist of the active signatures, when more and more active codes are added. This provides a measure of “distance” of SS∗ (the Gram matrix of S to an identity matrix). In essence it indicates how close the set is to being a WBE signature set since SS∗ = N m Im is a necessity for a WBE set with N users and dimension m [13]. Following [23, eq. (5.8.10)], it also gives some measure of separability of the signatures when using linear receivers such as the MMSE receiver (see also the discussion in [6]). ¯ ≥ m. We begin To make this comparison, we consider N with a baseline set of m sequences that are optimally selected such that the condition number of their Gram matrix is as small as possible. The idea behind this is that the system may still mainly designed for m users, thus it makes sense to select in advance codes for the most frequently used case. Then we successively add to each set a randomly chosen sequence from the remaining sequences until we reach the full 2mset of codes. In each step we compute the condition number of the sequence matrix and the squared Frobenius norm of the corresponding Gram matrix (i.e., the sum-correlation) for both sets. The latter is compared to the optimal value given by the Welch bound. The results are shown in Fig. 2-Fig. 3. When comparing the condition numbers in Fig. 2 we see that Grassmannian sequences clearly outperform WBE sequences and are relatively insensitive to the number of active codes. If a WBE set were designed for each subset ¯ | then the condition number would always be one. size |N The fact that Grassmannian sequences are not as good as redesigning a WBE for each possible number of users is offset by their flexibility in not having to continually reassign codes. Examining Fig. 3 shows that both sets have a similar distance to the ideal WBE set custom designed for the number of active signatures. Thus from the point-of-view of total correlation,
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interest to consider the equivalent of generalized WBE sequences [2] where it is expected that Grassmannian sequences will have a similar roll but only for the non-oversized users.
35 WBE MWBE 30
R EFERENCES
condition number
25
20
15
10
5
20
40
60 80 number of additional vectors
100
120
Fig. 2. Comparison of condition number of sequence matrices, when more and more users are successively added to the system.
500 WBE MWBE bound 450
total sum−squared correlation
400
350
300
250
200
150 20
40
60 80 number of additional vectors
100
120
Fig. 3. Comparison of the sum correlation of both sets, when more and more users are successively added to the system.
enforcing the maximally spaced equal angular property is no better or worse than an arbitrary WBE set. VII. C ONCLUSION AND F UTURE W ORK In this paper we studied an important subclass of WBE sequences known as Grassmannian sequences. We showed that this class of signatures has a pleasing interference invariance property for subsets of active users. While we considered sequences with arbitrary entries, it is of interest to study sequences from finite alphabets. It is unclear when equiangular sequence sets might exist in this case. Further, in this paper we only considered subsets of WBE sequences. It is also of
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[1] M. Rupf and J. L. Massey, “Optimum sequence multisets for synchronous code-division multiple-access channels,” IEEE Trans. Info. Theory, vol. 40, no. 4, pp. 1261–1266, 1994. [2] P. Viswanath and V. Anantharam, “Optimal sequences and sum capacity of synchronous CDMA systems,” IEEE Trans. Info. Theory, vol. 45, no. 6, pp. 1984–1991, Sept. 1999. [3] D. V. Sarwate, “Meeting the Welch bound with equality,” in Sequences and their applications (Singapore, 1998). London: Springer, 1999, pp. 79–102. [4] V. Kravcenko, H. Boche, F. Fitzek, and A. Wolisz, “No need for signaling: Investigation of capacity and quality of service for multi-code cdma systems usign the wbe ++ approach,” in Proc. 4th Int. Workshop on Mob. and Wireless Commun. Networks, 2002, pp. 110–114. [5] L. Welch, “Lower bounds on the maximum cross-correlation of signals,” IEEE Trans. Info. Theory, vol. 20, pp. 397–399, 1974. [6] T. Strohmer and R. W. Heath Jr., “Grassmannian frames with applications to coding and communications,” Applied and Computational Harmonic Analysis, vol. 14, no. 3, pp. 257–275, July 2003. [7] A. Shanbhag and J. Holtzman, “Optimal QPSK modulated quasiorthogonal functions for IS-2000,” in IEEE 6-th Int. Symp. Spread Spectrum Tech. & Appli., New Jersey, USA, Sept. 2000, pp. 756–760. [8] H. D. Schotten and H. Hadinejad-Mahram, “Analysis of a CDMA downlink with non-orthogonal spreading sequences for fading channels,” in Proc. of Veh. Tech. Conf., 2000, pp. 1782–1786. [9] K. Yang, Y.-K. Kim, and P. V. Kumar, “Quasi-orthogonal sequences for code-division multiple-access systems,” IEEE Trans. Info. Theory, vol. 46, no. 3, pp. 982–993, 2000. [10] S. Ulukus and R. D. Yates, “Iterative construction of optimum signature sequence sets in synchronous CDMA systems,” IEEE Trans. Info. Theory, vol. 47, no. 5, pp. 1989–1998, 2001. [11] J. A. Tropp, R. W. Heath Jr., and T. Strohmer, “An alternating projection algorithm for constructing quasi-orthogonal CDMA signature sequences,” in Proc. Int. Symp. on Info. Theo., 2003. [12] P. Viswanath, V. Anantharam, and D. N. C. Tse, “Optimal sequences, power control, and user capacity of synchronous CDMA systems with linear MMSE multiuser receivers,” IEEE Trans. Info. Theory, vol. 45, no. 6, pp. 1968–1983, 1999. [13] J. L. Massey and T. Mittelholzer, “Welch’s bound and sequence sets for code division multiple-access systems,” in Sequences II: Methods in Communication, Security, and Computer Science. New York: SpringerVerlag, 1993, pp. 63–78. [14] T. Strohmer, R. W. Heath Jr., and A. J. Paulraj, “On the design of optimal sequences for CDMA systems,” in Proceedings of the Thirty-Sixth Asil. Conf. on Sig., Sys. and Comp., vol. 2, Nov. 2002, pp. 1434–1438. [15] R. W. Heath Jr., T. Strohmer, and A. J. Paulraj, “On quasi-orthogonal signatures for CDMA systems,” submitted to IEEE Trans. Inform. Theory September 2002. [16] J. H. Conway, R. H. Hardin, and N. J. A. Sloane, “Packing lines, planes, etc.: packings in Grassmannian spaces,” Experiment. Math., vol. 5, no. 2, pp. 139–159, 1996. [17] A. R. Calderbank, R. H. Hardin, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “A group-theoretic framework for the construction of packings in Grassmannian spaces,” J. Algebraic Combin., vol. 9, no. 2, pp. 129– 140, 1999. [18] P. Delsarte, J. M. Goethals, and J. J. Seidel, “Bounds for systems of lines and Jacobi poynomials,” Philips Res. Repts, vol. 30, no. 3, pp. 91∗ –105∗ , 1975, issue in honour of C.J. Bouwkamp. [19] R. Rankin, “The closest packing of spherical caps in n dimensions,” Proc. Glasgow Math. Assoc., vol. 2, pp. 139–144, 1955. [20] M. Rosenfeld, “In praise of the Gram matrix,” in The mathematics of Paul Erd˝os, II. Berlin: Springer, 1997, pp. 318–323. [21] N. J. A. Sloane, “Packings in Grassmannian spaces,” http://www.research.att.com/ ˜njas/grass/index.html. [22] R. Paley, “On orthogonal matrices,” J. Math. Phys., vol. 12, pp. 311–320, 1933. [23] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge University Press, 1985.
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