Performance bounds and design criteria for estimating finite rate of innovation signals
Abstract: 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.