Bounds on Fast Decodability of Space-Time Block Codes, Skew-Hermitian Matrices, and Azumaya Algebras

Abstract

We study fast lattice decodability of space-time block codes for n transmit and receive antennas, written very generally as a linear combination Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2l</sup> s <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , where the si are real information symbols and the A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> are n×n ℝ-linearly independent complex-valued matrices. We show that the mutual orthogonality condition A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> * + A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> * = 0 for distinct basis matrices is not only sufficient but also necessary for fast decodability. We build on this to show that for full-rate (l = n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) transmission, the decoding complexity can be no better than |S|(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> +1), where |S| is the size of the effective real signal constellation. We also show that for full-rate transmission, g-group decodability, as defined by Jithamithra and Rajan, is impossible for any g ≥ 2. We then use the theory of Azumaya algebras to derive bounds on the maximum number of groups into which the basis matrices can be partitioned so that the matrices in different groups are mutually orthogonal-a key measure of fast decodability. We show that in general, this maximum number is of the order of only the 2-adic value of n. In the case where the matrices A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> arise from a division algebra, which is most desirable for diversity, we show that the maximum number of groups is only 4. As a result, the decoding complexity for this case is no better than |S|⌈l/2⌉ for any rate l.