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首页> 外文期刊>IEEE Transactions on Information Theory >Robust deconvolution of deterministic and random signals
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Robust deconvolution of deterministic and random signals

机译：确定性和随机信号的鲁棒反卷积

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摘要

We consider the problem of designing an estimation filter to recover a signal x[n] convolved with a linear time-invariant (LTI) filter h[n] and corrupted by additive noise. Our development treats the case in which the signal x[n] is deterministic and the case in which it is a stationary random process. Both formulations take advantage of some a priori knowledge on the class of underlying signals. In the deterministic setting, the signal is assumed to have bounded (weighted) energy; in the stochastic setting, the power spectra of the signal and noise are bounded at each frequency. The difficulty encountered in these estimation problems is that the mean-squared error (MSE) at the output of the estimation filter depends on the problem unknowns and therefore cannot be minimized. Beginning with the deterministic setting, we develop a minimax MSE estimation filter that minimizes the worst case point-wise MSE between the true signal x[n] and the estimated signal, over the class of bounded-norm inputs. We then establish that the MSE at the output of the minimax MSE filter is smaller than the MSE at the output of the conventional inverse filter, for all admissible signals. Next we treat the stochastic scenario, for which we propose a minimax regret estimation filter to deal with the power spectrum uncertainties. This filter is designed to minimize the worst case difference between the MSE in the presence of power spectrum uncertainties, and the MSE of the Wiener filter that knows the correct power spectra. The minimax regret filter takes the entire uncertainty interval into account, and as demonstrated through an example, can often lead to improved performance over traditional minimax MSE approaches for this problem.
机译：我们考虑设计一个估计滤波器以恢复与线性时不变（LTI）滤波器h [n]卷积并被加性噪声破坏的信号x [n]的问题。我们的开发处理了信号x [n]是确定性的情况和信号x [n]是平稳随机过程的情况。两种表述都利用了有关基础信号类别的先验知识。在确定性设置中，假定信号具有有界（加权）能量；在随机设置中，信号和噪声的功率谱在每个频率上都有界。这些估计问题中遇到的困难是，估计滤波器输出处的均方误差（MSE）取决于问题未知数，因此无法最小化。从确定性设置开始，我们开发了一个minimax MSE估计滤波器，该滤波器在有界范数输入类别上最小化真实信号x [n]与估计信号之间的最坏情况的逐点MSE。然后，对于所有允许的信号，我们确定在最小最大MSE滤波器的输出处的MSE小于在常规逆滤波器的输出处的MSE。接下来，我们处理随机情况，为此，我们提出了一个极大极小后悔估计滤波器来处理功率谱不确定性。设计该滤波器的目的是，在存在功率谱不确定性的情况下，最小化MSE与知道正确功率谱的Wiener滤波器的MSE之间的最坏情况差异。 minimax后悔滤波器考虑了整个不确定性区间，并且通过一个示例证明，与该问题相比，与传统的minimax MSE方法相比，通常可以提高性能。

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