Reconstruction of stationary and non-stationary signals by the generalized Prony method

Abstract

We employ the generalized Prony method in [T. Peter and G. Plonka, A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators, Inverse Problems 29 (2013) 025001] to derive new reconstruction schemes for a variety of sparse signal models using only a small number of signal measurements. By introducing generalized shift operators, we study the recovery of sparse trigonometric and hyperbolic functions as well as sparse expansions into Gaussians chirps and modulated Gaussian windows. Furthermore, we show how to reconstruct sparse polynomial expansions and sparse non-stationary signals with structured phase functions.

Reconstruction of Non‐Stationary Signals by the Generalized Prony Method

“…Theorem 2.1 Let S H,G,h be of the form (2) with H and G as in (1). Then S H,G,h possesses eigenfunctions of the form H(x) e αG(x) corresponding to the eigenvalue e αh for α ∈ R.…”

“…The generalized Prony method in [1,2,4] enables us to reconstruct sparse expansions of eigenfunctions of a linear operator. Therefore, we try to find a linear shift operator possessing eigenfunctions of the form H(x) e αj G(x) .…”

“…. , M see [2]. For the recovery of the c j and β j we use that cos (x + y) − cos (x − y) = −2 sin (x) sin (y) and that V = (sin (α j lh)) M −1,M l=0,j=1 is invertible for α j and h for j = 1, .…”

The Generalized Operator Based Prony Method

“…The sampling matrix in (3.8) applies the idea that instead of F k = F (ϕ(A) k ) we can also use Remark 3.13. A slightly different sampling scheme was applied in [21] and in [17], where the Prony polynomial has been written using a Chebyshev polynomial basis instead of the monomial basis.…”

“…Indeed, the shift operator S τ , with S τ f := f (· + τ), and its generalizations play a special role, since the power S ℓ τ is equivalent to S ℓτ , i.e., to a simple shift operator with shift length ℓτ. Expansions into eigenfunctions of generalized shift operators are therefore of special interest, since they can be recovered just by suitable function samples, see [17].…”