Envelope extraction in nuclear magnetic resonance (NMR) is a fundamental step for processing the data space generated by this technique. Envelope detection accuracy improves with increasing the number of sampling points; however, we propose a novel transform that enables acceptable envelope extraction with significantly fewer sampling points, even without meeting the Nyquist rate. In this paper, we challenge the traditional scale definition and demonstrate that classic scaling lacks a physical referent in all situations. To achieve this aim, we introduce a scale based on the variations of space-invariant states, rather than the observable characteristics of matter and energy. According to this definition of the scale, we distinguished two kinds of observers: scale-variant and scale-invariant. We demonstrated that converting a scale-variant observer to a scale-invariant observer is equivalent to envelop extraction. To analyse and study the theories presented in the paper, we have designed and implemented an Earth-field NMR setup and used real data generated by it to evaluate the performance of the proposed envelope-detection transform. We compared the output of the proposed transform with that of classic and state-of-the-art methods for parameter recovery of NMR signals.

翻译：核磁共振（NMR）中的包络提取是处理该技术生成数据空间的基本步骤。包络检测精度通常随采样点数量的增加而提高；然而，我们提出了一种新型变换，即使在不满足奈奎斯特采样率的情况下，也能以显著更少的采样点实现可接受的包络提取。本文挑战了传统的尺度定义，并证明经典尺度在所有情况下都缺乏物理参照。为实现这一目标，我们引入了一种基于空间不变态变化的尺度，而非基于物质与能量的可观测特性。根据这一定义，我们区分了两类观测者：尺度变异观测者与尺度不变观测者。我们证明了将尺度变异观测者转换为尺度不变观测者等价于包络提取。为分析和验证本文提出的理论，我们设计并搭建了一套地磁场核磁共振装置，并使用其产生的真实数据评估了所提包络检测变换的性能。我们将所提变换的输出与经典及前沿方法在核磁共振信号参数恢复方面进行了比较。