[论文解读] Van Vleck correction generalization for complex correlators with multilevel quantization

L. Benkevitch, A. E. E. Rogers+5 more|arXiv (Cornell University)|Aug 15, 2016

Statistical and numerical algorithms被引用 10

一句话总结

本文将射电干涉测量中数字相关器的范弗莱克校正推广至处理具有多级量化和不等信号标准差的复信号。提出一种从量化数据估计模拟信号标准差的方法，并利用广义模型反演量化相关响应，将默奇森宽场阵列（MWA）的模拟相关误差从约6%降低至<0.1%。

ABSTRACT

Remote sensing with phased antenna arrays is based on measurement of the cross-correlations between the signals from each antenna pair. Digital correlators have systematic errors due to the quantization losses. The correlation errors allow substantial abatement based on the assumption that the analog signals are stochastic processes sampled from a statistical distribution (usually the Gaussian). The correlation correction technique is named after Van Vleck who was the first to apply it to two-level clipping quantizers. The correction is especially important for high correlation levels, e.g. in studies of solar radio emissions. We offer a generalized method that for every antenna pair inputs the quantized signals' covariance and standard deviations, and outputs high-precision estimates of the analog correlation. Although correlation correction methods have been extensively investigated in the past, there are several problems that, as far as we know, have not been published yet. We consider a very general quantization scheme with arbitrary set of transition thresholds and output levels, and our correction method is designed for correlations obtained from signals with generally unequal standard deviations. We also provide a method for estimation of the analog standard deviation from the quantized one for subsequent use in the correlation correction. We apply the correction to the the complex-valued analytic signals, overwhelmingly used in modern remote sensing systems with arrays of antennas. The approach is valid not only for analytic signals with the imaginary part being the Hilbert transform of the real one, but also for more general, circularly symmetric complex processes whose real and imaginary parts may have arbitrary relationships to each other. This work was motivated by the need for greater precision in analysis of data from the Murchison Widefield Array (MWA).

研究动机与目标

• 解决射电干涉仪中复信号经多级量化后缺乏相关校正方法的问题。
• 解决从量化信号中估计标准差不准确的问题，该问题会导致相关校正产生偏差。
• 开发一种广义范弗莱克校正方法，可适用于任意量化等级和阈值，而不仅限于双级削波。
• 为具有不等标准差的信号实现高精度相关估计，这对太阳和太阳风观测至关重要。
• 通过校正高相关性区域中由量化引起的关联误差，提升默奇森宽场阵列（MWA）的成像动态范围。

提出的方法

• 利用具有任意阈值和输出电平的广义量化模型，将数字相关器响应表述为模拟相关性和信号标准差的函数。
• 通过反演量化相关方程（式20），利用测量得到的量化协方差和估计的模拟标准差，估计真实的模拟相关性。
• 推导出一个反向映射（式34），用于从量化标准差估计模拟信号标准差，以校正量化引入的偏差。
• 通过预计算的查找表实现校正，适用于规则量化模式（如4位MWA方案），支持实时应用。
• 将该方法应用于复解析信号，其中实部与虚部不一定是希尔伯特对，突破了经典范弗莱克假设的限制。
• 通过在信号标准差对与相关性值的网格上进行仿真，验证该方法在高相关性区域的表现。

实验结果

研究问题

• RQ1 如何将范弗莱克校正推广至具有任意阈值和输出电平的多级量化方案？
• RQ2 信号标准差不匹配对量化相关性有何影响？当两个信号均被量化时，如何校正该影响？
• RQ3 能否从量化数据中准确估计模拟信号标准差？此类估计的误差模型是什么？
• RQ4 不同信号标准差对下的逆相关函数形状（即ρ作为测量相关性的函数）如何变化？
• RQ5 该广义校正方法在默奇森宽场阵列等实际系统中，能在多大程度上降低相关误差？

主要发现

• 广义范弗莱克校正可将4位默奇森宽场阵列（MWA）量化器的典型相关误差从约6%降低至±0.1%以内。
• 在高相关性区域（|ρ| ≈ 1），仅当两信号标准差相等时，逆校正函数才为凸性；否则在上端通常呈凹性。
• 在(σx, σy)平面上的对角线（σx = σy）是唯一所有校正函数在ρ = 1处均为凸性的区域，表明对称信号具有独特行为。
• 该方法可通过式（28）的逆运算，从量化数据中准确估计模拟标准差，这对实现无偏相关校正至关重要。
• 仿真结果表明，在典型条件下，校正后的残余误差低至±0.01%–0.001%，证实了对强且高度相关信号的高精度。
• 即使存在异常值，该校正方法仍有效，但若异常值比例过高，性能会略有下降，凸显了数据质量控制的重要性。