标题： 克朗尼格-克拉默斯，博德和零的含义 显示英文标题

标题： Kramers-Kronig, Bode, and the meaning of zero

摘要： 因果关系的含义，通过线性响应函数实部和虚部之间的Kramers-Kronig关系来体现，是物理课程中熟悉的内容。 1937年，Bode推导出了响应增益（幅度）和相位之间的类似关系。 尽管Kramers-Kronig关系是一个等式，但Bode的关系实际上是一个不等式。 这种令人惊讶的差异可以通过简单的例子加以解释，并最终追溯到由物理物体和测量装置组成的系统中信息流动的延迟。 显示英文摘要

摘要： The implications of causality, as captured by the Kramers-Kronig relations between the real and imaginary parts of a linear response function, are familiar parts of the physics curriculum. In 1937, Bode derived a similar relation between the magnitude (response gain) and phase. Although the Kramers-Kronig relations are an equality, Bode's relation is effectively an inequality. This perhaps-surprising difference is explained using elementary examples and ultimately traces back to delays in the flow of information within the system formed by the physical object and measurement apparatus.