Performance bounds and design criteria for estimating finite rate of innovation signals
IEEE Transactions on Information Theory, vol. 58 (2012), pp. 4993-5015
Zvika Ben-Haim, Tomer Michaeli, Yonina C. Eldar
In this paper, we consider the problem of estimating ﬁnite rate of innovation (FRI) signals from noisy measurements, and speciﬁcally analyze the interaction between FRI techniques and the underlying sampling methods. We ﬁrst obtain a fundamental limit on the estimation accuracy attainable regardless of the sampling method. Next, we provide a bound on the performance achievable using any speciﬁc sampling approach. Essential differences between the noisy and noise-free cases arise from this analysis. In particular, we identify settings in which noise-free recovery techniques deteriorate substantially under slight noise levels, thus quantifying the numerical instability inherent in such methods. This instability, which is only present in some families of FRI signals, is shown to be related to a speciﬁc type of structure, which can be characterized by viewing the signal model as a union of subspaces. Finally, we develop a methodology for choosing the optimal sampling kernels for linear reconstruction, based on a generalization of the Karhunen–Loeve transform. The results are illustrated for several types of time-delay estimation problems.