Several current ultra-wide band applications, such as millimeter wave radar and communication systems, require high sampling rates and therefore expensive and energy-hungry analogto-digital converters (ADCs). In applications where cost and power constraints exist, the use of high-precision ADCs is not feasible and the designer must resort to ADCs with coarse quantization. Consequently the interest in the topic of signal parameter estimation from quantized data has increased significantly in recent years. The Cramer-Rao bound (CRB) is an important yardstick in any parameter estimation problem. Indeed it lower bounds the variance of any unbiased parameter estimator. Moreover, the CRB is an achievable limit, for instance it is asymptotically attained by the maximum likelihood estimator (under regularity conditions), and thus it is a useful benchmark to which the accuracy of any parameter estimator can and should be compared. A formula for the CRB for signal parameter estimation from real-valued quantized data has been presented in but its derivation was somewhat sketchy. The said CRB formula has been extended for instance in to complex-valued quantized data, but again its derivation was rather sketchy. The special case of binary (1-bit) ADCs and a signal consisting of one sinusoid has been thoroughly analyzed in . The CRB formula for a binary ADC and a general real-valued signal has been derived.

中文翻译：

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从量化数据估计信号参数的 Cramer-Rao 界

当前的几种超宽带应用，例如毫米波雷达和通信系统，需要高采样率，因此需要昂贵且耗能的模数转换器 (ADC)。在存在成本和功耗限制的应用中，使用高精度 ADC 是不可行的，设计人员必须求助于粗量化的 ADC。因此，近年来，人们对量化数据中的信号参数估计这一主题的兴趣显着增加。Cramer-Rao 界 (CRB) 是任何参数估计问题的重要尺度。事实上，它降低了任何无偏参数估计器的方差。此外，CRB 是一个可实现的限制，例如，它是由最大似然估计量（在正则条件下）渐近获得的，因此，它是一个有用的基准，任何参数估计器的准确性都可以而且应该与之进行比较。已经提出了用于从实值量化数据估计信号参数的 CRB 公式，但它的推导有些粗略。上述 CRB 公式已经扩展到例如复值量化数据，但它的推导还是相当粗略的。二进制（1 位）ADC 和由一个正弦波组成的信号的特殊情况已在 . 已推导出二进制 ADC 和一般实值信号的 CRB 公式。上述 CRB 公式已经扩展到例如复值量化数据，但它的推导还是相当粗略的。二进制（1 位）ADC 和由一个正弦波组成的信号的特殊情况已在 . 已推导出二进制 ADC 和一般实值信号的 CRB 公式。上述 CRB 公式已经扩展到例如复值量化数据，但它的推导还是相当粗略的。二进制（1 位）ADC 和由一个正弦波组成的信号的特殊情况已在 . 已推导出二进制 ADC 和一般实值信号的 CRB 公式。