Stationary Processes forming Frames and Basis: Some Applications

JUAN MIGUEL MEDINA; BRUNO CERNUSCHI FRÍAS

Mar del Plata

Otro; Reunión Anual de la Unión Matemática Argentina UMA 2009; 2009

Unión Matemática Argentina

We will give necessary and sufficient conditions for {Xjk}j=1...nk2Z the samples taken at uniform rate from a wide sense stationary processes (or an “observed”processes) to form a frame or a Riesz Basis of its span. In some way this is an anologue problem to that of finding conditions for frames of a shift invariant subspace of L2(R) with a finite number of generators.  Moreover it is also possible to give conditions for the dual frames. This is be very useful to sample and reconstruct continuous time random processes - signals as in the deterministic case of L2(R) functions, and as in the latter case it is easy to prove this provides a “robust” representation of the signal if the samples are corrupted with additive noise.{Xjk}j=1...nk2Z the samples taken at uniform rate from a wide sense stationary processes (or an “observed”processes) to form a frame or a Riesz Basis of its span. In some way this is an anologue problem to that of finding conditions for frames of a shift invariant subspace of L2(R) with a finite number of generators.  Moreover it is also possible to give conditions for the dual frames. This is be very useful to sample and reconstruct continuous time random processes - signals as in the deterministic case of L2(R) functions, and as in the latter case it is easy to prove this provides a “robust” representation of the signal if the samples are corrupted with additive noise.L2(R) with a finite number of generators.  Moreover it is also possible to give conditions for the dual frames. This is be very useful to sample and reconstruct continuous time random processes - signals as in the deterministic case of L2(R) functions, and as in the latter case it is easy to prove this provides a “robust” representation of the signal if the samples are corrupted with additive noise. Moreover it is also possible to give conditions for the dual frames. This is be very useful to sample and reconstruct continuous time random processes - signals as in the deterministic case of L2(R) functions, and as in the latter case it is easy to prove this provides a “robust” representation of the signal if the samples are corrupted with additive noise.L2(R) functions, and as in the latter case it is easy to prove this provides a “robust” representation of the signal if the samples are corrupted with additive noise.