Fast algorithms for updating signal subspaces

In various real-time signal processing and communication applications, it is often required to track a low-dimensional signal subspace that slowly varies with time. Conventional methods of updating the signal subspace rely on eigendecomposition or singular value decomposition, which is computationally expensive and difficult to implement in parallel. Recently, Xu and Kailath proposed fast and parallelizable Lanczos-based algorithms for estimating the signal subspace based on the data matrices or the covariance matrices. In this paper, we shall extend these algorithms to achieve fast tracking of the signal subspace. The computational complexity of the new methods is O(M ²d) per update, where M is the size of the data vectors and d is the dimension of the signal subspace. Unlike most tracking methods that assume d is fixed and/or known a priori, the new methods also update the signal subspace dimension. More importantly, under certain stationarity conditions, we can show that the Lanczos-based methods are asymptotically equivalent to the more costly SVD or eigendecomposition based methods and that the estimation of d is strongly consistent. Knowledge of the previous signal subspace estimate is incorporated to achieve better numerical properties for the current signal subspace estimate. Numerical simulations for some signal scenarios are also presented