Signature Design for CDMA Systems

Recently there has been considerable interest in optimum signature sequences for CDMA (code division multiple access) communication systems. In brief, CDMA systems assign signatures to different users for the purpose of allowing multiple users to simultaneously access the channel. The problem of finding optimum signatures in terms of capacity, it turns out, relates to an equivalent problem of constructing a certain correlation matrix with given Eigenvalues. This problem in turn relates to the problem of frame design.

We have explored the problem of signature design in a number of different ways. Our initial work was to propose and examine special subclasses of optimum signatures that have a lot of structure. We proposed two different signatures sets. One we called multiple orthonormal basis (multi-ONB) signature sets. They involve concatenating multiple orthonormal bases that are as “different” as possible. The other we called Grassmannian signature sets. Grassmannian signatures arise from the connection between line packing and the signature construction problem.

Our initial results are summarized in:

R. W. Heath, Jr. and T. Strohmer, “On Quasi-Orthogonal Signatures for CDMA Systems,” in *Proc. of *Allerton Conf. on Comm. Control and Comp., Oct. 3 – Oct. 5, 2002.

T. Strohmer, R. W. Heath, Jr., and A. J. Paulraj “On The Design Of Optimal Sequences For CDMA Systems,” in *Proc. of **IEEE Asilomar Conf. on Signals, Systems, and Computers*, vol. 2, pp. 1434-1438, Pacific Grove, California, 2002.

More details can be found in:

R. W. Heath, Jr., T. Strohmer, and A. J. Paulraj, “On Quasi-Orthogonal Signatures for CDMA Systems,” submitted to the *IEEE Trans. on Info. Theory*, September 2002.

We have also more extensively investigated some of the benefits of Grassmannian signature sets. In particular, we argue that they are good signatures to use in practical systems when users are coming and going due to an *interference invariance *property that arises from the equiangular property. More results on this topic are available in the following:

R. W. Heath, Jr., T. Strohmer, and Arogyaswami J. Paulraj, “Grassmannian Signatures for CDMA Systems,” to appear in *Proc. of IEEE Global Telecommunications Conf. *, San Francisco, CA, Dec. 1-5, 2003.

As alluded to earlier, the problem of signature design can be related to an equivalent problem of matrix design. In particular, it can be shown that this relates to an inverse eigenvalue problem. We have derived a alternating projection algorithm that involves alternating between a closed set and a convex set.

Our initial results are summarized in:

J. A. Tropp, R. W. Heath, Jr., and T. Strohmer, “Optimal CDMA Signature Sequences, Inverse Eigenvalue Problems And Alternating Minimization” *Proc. of the* International Symposium on Information Theory, p. 407, Pacifico YOKOHAMA, Yokohama, JAPAN June 29 – July 4, 2003.

Relationships with certain finite-step algorithms is made in:

J. A. Tropp, I. Dhillon, and R. W. Heath, Jr., “Finite-Step Algorithms for Constructing Optimal CDMA Signature Sequences,” Wireless Networking and Communications Group Technical Report WNCG-TR-2003-05-08, also submitted to the *IEEE Trans. on Info. Theory*, May 2003, revised November 2003.

An important application of this algorithm is to finding constrained signatures. We have studied the problem of finding optimal signatures with certain peak-to-average (PAR) constraints in the following:

J. A. Tropp, I. Dhillon, R. W. Heath, Jr., and T. Strohmer, “CDMA Signature Sequences With Low Peak-To-Average Ratio Via Alternating Minimization,” to appear in *Proc. of **IEEE Asilomar Conf. on Signals, Systems, and Computers*, Pacific Grove, CA, USA, Nov. 9-12.

A more complete discussion of the alternating projection algorithm is available in the following which uses it to construct optimum signatures, Grassmannian signatures, and PAR constrained signatures (contains the proofs for the above paper).

J. A. Tropp, I. Dhillon, R. W. Heath, Jr., T. Strohmer “Designing Structured Tight Frames Via An Alternating Projection Method,” The University of Texas at Austin, ICES Report 03-50, December 2003, also submitted to the *IEEE Trans. on Info. Theory*.

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