从大偏差角度研究最优参数调制估计
On optimum parameter modulation-estimation from a large deviations perspective
摘要：
我们考虑的问题，联合最佳调制和估计的实值随机参数，通过加性高斯白噪声（AWGN）信道，其中的性能指标是大偏差的估计行为，即，指数衰减率（作为观测时间的函数）的概率估计误差将超过一定的阈值。我们的基本结果是在提供最快的可实现的指数衰减率的精确表征，在所有可能的调制器估计器（发射机-接收机）对，其中调制器仅限于信号功率，但不限于带宽。该指数速率由AWGN信道的可靠性函数给出。我们还讨论了实现这种最佳性能的几种方法，其中之一是基于参数的量化，然后是最佳信道编码和调制，如果从联合信源-信道编码的角度来看，这会产生基于分离的发射机。这是尽管事实上，在一般情况下，当考虑误差指数，源通道分离定理不成立。我们还讨论了几个观察，修改和扩展这个结果在几个方向，包括其他渠道，和多维参数向量的情况下。关于后者，我们的发现之一是，在参数向量的维度中存在突然的阈值效应：在某个临界维度以下，过度估计误差的概率仍可能呈指数衰减，但超过该值，它必须收敛到1。
英文摘要：
We consider the problem of jointly optimum modulation and estimation of a real-valued random parameter, conveyed over an additive white Gaussian noise (AWGN) channel, where the performance metric is the large deviations behavior of the estimator, namely, the exponential decay rate (as a function of the observation time) of the probability that the estimation error would exceed a certain threshold. Our basic result is in providing an exact characterization of the fastest achievable exponential decay rate, among all possible modulator-estimator (transmitter-receiver) pairs, where the modulator is limited only in the signal power, but not in bandwidth. This exponential rate turns out to be given by the reliability function of the AWGN channel. We also discuss several ways to achieve this optimum performance, and one of them is based on quantization of the parameter, followed by optimum channel coding and modulation, which gives rise to a separation-based transmitter, if one views this setting from the perspective of joint source-channel coding. This is in spite of the fact that, in general, when error exponents are considered, the source-channel separation theorem does not hold true. We also discuss several observations, modifications and extensions of this result in several directions, including other channels, and the case of multidimensional parameter vectors. One of our findings concerning the latter, is that there is an abrupt threshold effect in the dimensionality of the parameter vector: below a certain critical dimension, the probability of excess estimation error may still decay exponentially, but beyond this value, it must converge to unity.
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