Transmission Subspace Tracking for MIMO Systems With Low-Rate Feedback

This paper describes a feedback algorithm for tracking the dominant subspaces of continuously time-varying channels in multiantenna communication systems. The nature of the problem is quantization of subspaces. It is well known that subspaces can be mathematically modeled as points in a Grassmann manifold. We model the variations between the dominant subspaces of channels at adjacent time instants to be along geodesics in the Grassmann manifold. Instead of quantizing the subspaces themselves, we propose to quantize the geodesic trajectory connecting two subspaces. More specifically, we quantize a key entity that characterizes a geodesic arc: the velocity matrix, which resembles angular speed in a one-dimensional complex space. Two techniques are proposed for quantizing the velocity matrix of the geodesic. In the first, a 1-bit feedback is utilized to indicate the preferred sign of a random velocity matrix of the geodesic. In the other, the velocity matrix is quantized using a Gaussian vector quantization codebook. Numerical results show that the performance of the proposed 1-bit feedback algorithm is better than a previously proposed Grassmannian subspace packing scheme at low-to-medium Doppler frequencies and better than a gradient sign feedback scheme at all Doppler frequencies. In our simulations, the Gaussian vector quantization algorithm is always better than the 1-bit feedback algorithm.