Abstract-In this paper, we develop soft decision equalization (SDE) techniques for frequency selective multiple-input multiple-output (MIMO) channels in the quest for low-complexity equalizers with error performance competitive to that of maximum likelihood (ML) sequence detection. We demonstrate that decision feedback equalization (DFE) based on soft-decisions, expressed via the posterior probabilities associated with feedback symbols, is able to outperform hard-decision DFE, with a low computational cost that is polynomial in the number of symbols to be recovered and linear in the signal constellation size. Building on the probabilistic data association (PDA) multiuser detector, we present two new MIMO equalization solutions to handle the distinctive channel memory. The first SDE algorithm adopts a zero-padded transmission structure to convert the challenging sequence detection problem into a block-by-block least-square formulation. It introduces key enhancement to the original PDA to enable applications in rank-deficient channels and for higher level modulations. The second SDE algorithm takes advantage of the Toeplitz channel matrix structure embodied in an equalization problem. It processes the data samples through a series of overlapping sliding windows to reduce complexity and, at the same time, performs implicit noise tracking to maintain near-optimum performance. With their low complexity, simple implementations, and impressive near-optimum performance offered by iterative soft-decision processing, the proposed SDE methods are attractive candidates to deliver efficient reception solutions to practical high-capacity MIMO systems. Simulation comparisons of our SDE methods with minimum-mean-square error (MMSE)-based MIMO DFE and sphere decoded quasi-ML detection are presented.Index Terms-Efficient reception algorithms, equalization for frequency selective MIMO channels, overlapping sliding windowing, soft decision.

For each n 1, we define an algebra having many properties that one might expect to hold for a Brauer algebra of type B n . It is defined by means of a presentation by generators and relations. We show that this algebra is a subalgebra of the Brauer algebra of type D nC1 and point out a cellular structure in it. This work is a natural sequel to the introduction of Brauer algebras of type C n , which are subalgebras of classical Brauer algebras of type A 2n 1 and differ from the current ones for n > 2. A novel feature is the failure of admissible root sets to describe the tops and bottoms of the diagrams corresponding to monomials in the Brauer algebra of type B n ; instead of these sets we use extended admissible sets in order to find normal forms for monomials in the algebra.

In this paper, we will define the Brauer algebras of Weyl types, and describe some propositions of these algebras. Especially, we prove the result of type G 2 to accomplish our project of Brauer algebras of nonsimply laced types.

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