The implications of causality are captured by the Kramers–Kronig relations between the real and imaginary parts of a linear response function. In 1937, Bode derived a similar relation between the magnitude (response gain) and the phase. Although the Kramers–Kronig relations are an equality, the Bode’s relation is effectively an inequality. This difference is explained using elementary examples and is traced back to delays in the flow of information within the system formed by the physical object and the measurement apparatus.
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The engineering literature uses the complex s plane and Laplace transform instead of the complex ω plane and Fourier transform. Because the two are rotated by 90°, our discussion of the analyticity properties in the lower and upper ω planes is equivalent to discussions in the engineering literature of analyticity properties of the left-hand and right-hand s planes.
17.
In the engineering literature, Eq. (10) usually has the opposite sign. This difference traces back to the engineers’ use of e−iωt rather than e+iωt in the forward Fourier transform.
18.
In formulating the gain-phase relation, we assume that the DC gain (that is, at ω = 0) is positive. A negative DC gain can be regarded as an overall conversion factor between input and output rather than as an extra 180° phase shift. Thus, for our purposes, both G(ω) = −1/(1 – iω) and 1/(1 – iω) have the same phase response.
19.
We also need to assume that G(ω) has no poles in the upper half of the complex ω-plane. Such poles correspond to unstable, exponentially growing motion and also add to the phase delay of the response. Only active systems, with external energy injection, can have such poles.
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2011
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