ITR: Adaptive Unitary Matrices For High-Speed Communications

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

A fundamental problem in a wide variety of wideband communications applications is that of adapting a unitary matrix. For example, consider an array-to-array AWGN wireless communications link in which the transmitter has knowledge of the channel. To achieve the Shannon capacity of such a link, the optimal transmitter prefilters the vector of transmitted symbols by the right unitary factor in the channel singularvalue decomposition. This prefilter, coupled with a matrix-valued matched filter at the receiver, converts the matrix channel into an equivalent bank of independent scalar channels, greatly simplifying the modulation, coding, and other signal processing tasks needed to maximize throughput. Despite the optimality of the unitary prefilter, however, the high complexity of the SVD (especially for rapidly varying channels) makes it difficult to realize in practice. A low-complexity algorithm for adapting a unitary matrix could be the catalyst that brings such optimal space-time processing closer to reality. Other applications requiring an adaptive unitary matrix include blind fractionally spaced equalization; blind multiuser detection for CDMA; blind combining for narrowband arrays; and signal-noise subspace separation and subspace tracking. The objective of this proposal is to develop low complexity and robust strategies for adapting unitary matrices, to assess their performance, convergence properties, and complexity, and to compare them to existing alternatives. The proposal is built around three core algorithms for adapting a unitary matrix U k T he MPLL algorithm, the subspace separator, and the adaptive SVD algorithm. All three take the form: U k + 1 = U kR l {g(z k ) z k }, where z k = U k * y k and y k is the receiver observation vector, and where R l {x z} denotes a particular unitary matrix that rotates x / || x || a fraction l of the way to z / || z ||. The memoryless function g(z) is anonlinear decision device for the MPLL, while it is a linear matrix for the subspace separator and the adaptive SVD algorithm. This recursion has a number of nice properties; it is conceptually simple, and simple algorithms are often the most robust; it ensures that U k is always unitary, since R l is unitary; there exists a lower complexity implementation that does not require full-rank matrix multiplication; and the recursion has a useful geometric interpretation that facilitates its application to new areas. Furthermore, as discussed in this proposal, preliminary results indicate that the recursion has excellent convergence properties. The inspiration for the proposed algorithms came from the scalar phase-locked loop (PLL). The PLL is clearly the workhorse for adapting unitary scalars, primarily because of its robustness, low complexity,good performance. It stands to reason, therefore, that some of these properties might carry over into the higher-dimensional problem of adapting unitary matrices. The proposed research constitutes a significant advancement in the theory of signal processing and its application to communication theory, and is applicable to a wide variety of signal processing applications beyond communications. The project will provide two Ph.D. students with extensive physical-layer training for next-generation wideband communications systems.