Hilbert space array methods for finite rank process estimation and ladder realizations for adaptive signal processing

We present a Hilbert space array approach for deriving fast estimation and adaptive signal processing algorithms, that are recursive in time and order. Ladder (or lattice) forms turn out to be the natural realizations of these algorithms. From a stochastic point of view, the natural class of processes associated with these techniques include stationary but also nonstationary processes of finite (displacement) rank, also referred to as shift-low-rank or alpha-stationary processes. They are encountered in adaptive signal processing, speech modeling and encoding, digital communication, radar and sonar, high-resolution spectral estimation, distance measures etc. The use of projections and orthonormalizations, e.g. via Gram Schmidt procedures, induces real and complex rotations as basic operations, resulting in magnitude normalized variables and numerically stable computations.