Reconstruction of Finite Rate of Innovation Signals with Model-Fitting Approach

Abstract

Abstract-Finite rate of innovation (FRI) is a recent framework for sampling and reconstruction of a large class of parametric signals that are characterized by finite number of innovations (parameters) per unit interval. In the absence of noise, exact recovery of FRI signals has been demonstrated. In the noisy scenario, there exist techniques to deal with non-ideal measurements. Yet, the accuracy and resiliency to noise and model mismatch are still challenging problems for real-world applications. We address the reconstruction of FRI signals, specifically a stream of Diracs, from few signal samples degraded by noise and we propose a new FRI reconstruction method that is based on a model-fitting approach related to the structured-TLS problem. The model-fitting method is based on minimizing the training error, that is, the error between the computed and the recovered moments (i.e., the FRI-samples of the signal), subject to an annihilation system. We present our framework for three different constraints of the annihilation system. Moreover, we propose a model order selection framework to determine the innovation rate of the signal; i.e., the number of Diracs by estimating the noise level through the training error curve. We compare the performance of the model-fitting approach with known FRI reconstruction algorithms and Cramér-Rao's lower bound (CRLB) to validate these contributions.Index Terms-Annihilating Filter, Cadzow, Cramér-Rao's lower bound (CRLB), finite-rate-of-innovation, iterative quadratic maximum likelihood (IQML), Kumaresan-Tufts, matrix pencil, model fitting, noise, reconstruction, sampling, structured total least squares (STLS), total least squares (TLS).

Sparse signal recovery using structured total maximum likelihood

“…More recent work [24] has advocated the use of methods based on FRI theory [9], [10], [19], [25], [26] as an alternative to solve the sparsity recovery problem.…”

“…In [19], FRI signal recovery is formulated as a model fitting problem using structured total least squares (STLS) [21]: …”

“…Recently, a model fitting based algorithm [19] tries to recover the FRI signal using structured total least squares (STLS) [20], [21]. The FRI model fitting problem is formulated as minimizing the residual in the noisy observation with a low rank constraint represented by an annihilation relation.…”

“…With an additional log-det term, the STML becomes more difficult to solve. Inspired by the linear constraint proposed in [19], we transform the STML formulation with the log-det term into a form similar to that of STLS. Using an iterative algorithm similar to the iterative quadratic maximum likelihood (IQML) [20] method, the sparse signal can be effectively estimated using our proposed STML method.…”

“…Section 3 reviews STLS approach [19] for FRI signal recovery. Section 4 introduces our proposed sparse signal recovery with STML and a novel iterative algorithm.…”