Wide Sense Stationary Processes Forming Frames

Frame sequences and representations for samplable random processes

“…Moreover, the dual basis is given by { e inλ (φ(λ)) −1 √ 2π } n∈Z , and then S n (t) = E(x t z n ) = sin(π(t−n)) π(t−n) , and as μ is equivalent to the restriction of the ordinary Lebesgue measure on [−π, π], one may take the operator T defined by the kernel: k(t, λ) = e itλ 1 [−π,π] (λ). Finally, we recall that in this case, the condition on μ of being absolutely continuous with respect to the Lebesgue measure is necessary, as it was proved in [19].…”

“…processes. From the stochastic version of the WSK theorem, under additional conditions, we obtain a Riesz basis or a frame sequence of samples which span the Hilbert space spanned by the whole process [19]. The representation of signals using frames has many practical applications [8], in particular, dealing with additive noise.…”

On the Papoulis Sampling Theorem: Some General Conditions

“…The following result by the authors [19] gives a necessary and sufficient condition for a vector stationary sequence to be a frame or a Riesz basis of the closed linear span of its scalar values. In [20] a similar result is proved for scalar sequences.…”

Frame sequences and representations for samplable random processes

“…processes. The stochastic version of the WSK theorem, under additional conditions, gives an orthogonal set or a Riesz basis of samples which spans the Hilbert space spanned by the whole process [13]. The representation of signals using Riesz basis has many practical applications [2].…”

“…In this case, the condition on µ of being absolutely continuous with respect to the Lebesgue measure, is necessary, as it was proved in[13].…”