高斯信道中的信息、估计和前瞻
Information, Estimation, and Lookahead in the Gaussian channel
摘要：
本文研究了被加性高斯白噪声污染的连续时间信号的先行均方估计问题。我们表明，互信息率函数，即，作为信噪比（SNR）的函数的互信息率通常不确定具有固定有限前视的最小均方误差（MMSE），这与先前在文献中建立的分别具有0和无限前视（滤波和平滑误差）的特殊情况相反。我们还建立了一个新的期望身份下的广义观测模型，高斯信道有一个SNR跳跃在$t=0$，捕捉前瞻和SNR之间的权衡。进一步研究了一类连续时间平稳高斯-马尔可夫过程（Ornstein-Uhlenbeck过程）作为信道输入，并在有限前视和信噪比（SNR）条件下，明确刻画了最小均方误差（MMSE）的行为.具有前瞻性的MMSE示出以指数方式快速收敛到非因果误差，其中指数是非因果误差的倒数。我们扩展我们的结果的混合物的Ornstein-Uhlenbeck过程，并使用所获得的洞察力，目前的MMSE的下限和上限与前瞻一类平稳高斯输入过程，其频谱可以表示为混合物的Ornstein-Uhlenbeck频谱。
英文摘要：
We consider mean squared estimation with lookahead of a continuous-time signal corrupted by additive white Gaussian noise. We show that the mutual information rate function, i.e., the mutual information rate as function of the signal-to-noise ratio (SNR), does not, in general, determine the minimum mean squared error (MMSE) with fixed finite lookahead, in contrast to the special cases with 0 and infinite lookahead (filtering and smoothing errors), respectively, which were previously established in the literature. We also establish a new expectation identity under a generalized observation model where the Gaussian channel has an SNR jump at $t=0$, capturing the tradeoff between lookahead and SNR. Further, we study the class of continuous-time stationary Gauss-Markov processes (Ornstein-Uhlenbeck processes) as channel inputs, and explicitly characterize the behavior of the minimum mean squared error (MMSE) with finite lookahead and signal-to-noise ratio (SNR). The MMSE with lookahead is shown to converge exponentially rapidly to the non-causal error, with the exponent being the reciprocal of the non-causal error. We extend our results to mixtures of Ornstein-Uhlenbeck processes, and use the insight gained to present lower and upper bounds on the MMSE with lookahead for a class of stationary Gaussian input processes, whose spectrum can be expressed as a mixture of Ornstein-Uhlenbeck spectra.
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