On the theory of random decrement

Abstract

In this paper, we will give a short overview of the theory of the random decrement (RD) technique invented around 1970 by Henry Cole for structural response analysis, and introduced in the seventies as part of the Ibrahim time domain identification technique. The RD technique introduced by Cole and Ibrahim was an algorithm with uncertain meaning until Vandiver et al. and later Brincker et al. presented the theory of the RD functions as conditional expectations of the responses. Since the potential of seeing the RD functions as a conditional expectation, seem to remain un-released after 30–40 years of their ideas, we want in this paper to give an overview of the major findings. First, we will present a general framework for the theory of RD, and then we will use this to re-introduce the work of Vandiver in a simple way to obtain the theoretical solution for the level triggering condition for a single time series. We then generalize Vandiver’s result to the cross RD case where we are averaging on one time series and applying the condition to another time series. For this case, we obtain the general result of the RD function being proportional to the cross correlation function between the two considered signals. Finally, we consider the general triggering condition, where we are conditioning on both the signal itself and on its derivative, to obtain the so-called fundamental solution, where the RD function turns out to be a linear combination of the cross correlation function and its derivative. Finally, we will discuss different application issues, and illustrate how the fundamental solution can be used for identification taking advantage of the extended information from the derivative of the cross correlation function.