In various real-time signal processing and communicationnapplications, it is often required to track a low-dimensional signalnsubspace that slowly varies with time. Conventional methods of updatingnthe signal subspace rely on eigendecomposition or singular valuendecomposition, which is computationally expensive and difficult tonimplement in parallel. Recently, Xu and Kailath proposed fast andnparallelizable Lanczos-based algorithms for estimating the signalnsubspace based on the data matrices or the covariance matrices. In thisnpaper, we shall extend these algorithms to achieve fast tracking of thensignal subspace. The computational complexity of the new methods is O(Mn2d) per update, where M is the size of the data vectors and dnis the dimension of the signal subspace. Unlike most tracking methodsnthat assume d is fixed and/or known a priori, the new methods alsonupdate the signal subspace dimension. More importantly, under certainnstationarity conditions, we can show that the Lanczos-based methods arenasymptotically equivalent to the more costly SVD or eigendecompositionnbased methods and that the estimation of d is strongly consistent.nKnowledge of the previous signal subspace estimate is incorporated tonachieve better numerical properties for the current signal subspacenestimate. Numerical simulations for some signal scenarios are alsonpresented
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