Bounding Rate and Performance of Differential Unitary Space-Time Codes Using Integral Geometry Measures

We consider lower bounding the maximum achievable rate of a differential unitary spacetime transmission subject to a prescribed performance. We characterize the performance using a suitably defined distance between two unitary matrices, which was previously obtained through pairwise error probability analysis. Considering the set of all unitary or orthogonal matrices as a compact manifold with some invariant volume form or kinematic density, we derive a Varshamov-Gilbert type lower bound on the transmission rate, which asserts the existence of a differential unitary space-time code with the given minimum diversity product for certain rates. On the way to obtaining this bound, the probability density function and the cumulative distribution function for the distance between a random unitary matrix and a fixed unitary matrix are also obtained in closed form in terms of Meijer G-functions.