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Performance Bounds and Design Criteria for Estimating Finite Rate of Innovation Signals

机译：估算创新信号有限速率的性能界限和设计标准

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摘要

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–Loève transform. The results are illustrated for several types of time-delay estimation problems.
机译：在本文中，我们考虑了从噪声测量中估算有限创新速率（FRI）信号的问题，并专门分析了FRI技术与基础采样方法之间的相互作用。无论采样方法如何，我们首先都会获得可达到的估计精度的基本限制。接下来，我们对使用任何特定采样方法可获得的性能进行了限制。噪声和无噪声情况之间的本质区别在于此分析。尤其是，我们确定了无噪音恢复技术在轻微的噪音水平下会严重恶化的设置，从而量化了此类方法固有的数值不稳定性。这种不稳定性（仅在某些FRI信号系列中才会出现）与特定类型的结构有关，可以通过将信号模型视为子空间的并集来表征。最后，我们基于Karhunen-Loève变换的推广，开发了一种方法，用于选择用于线性重构的最佳采样内核。说明了几种类型的时间延迟估计问题的结果。

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