The Kramers-Kronig in an Expectation Sense

A second issue is that the integral equations do not account explicitly for the stochastic character of experimental data. This requires solution of the Kramers-Kronig relations in an expectation sense, as discussed in the following section. The methods used to address the incomplete sampled frequency range will be described in a subsequent section. 

The contribution of stochastic error e co) to the observed value of the impedance at any given frequency co can be expressed as 

At any frequency co the expectation of the observed impedance E (Zob(it ))/ defined in equation (3.1), is equal to the value consistent with the Kramers-Kronig relations, i.e., 

Equations (22.71) and (22.72) are satisfied for errors that are stochastic and do not include the effects of bias. 

Equation (22.43) can be applied to obtain the imaginary part from the real part of the impedance spectrum only in an expectation sense, i.e.. 

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