Choose Go for a delta-correlated approximation of a stochastic process with autocovariance function GDD(τ)=Ae^-c|τ|

Question

Choose [latex] G_{0} [/latex] for a delta-correlated approximation of a stochastic process with autocovariance function

[latex] G_{FF}( au )=Ae^{-c\left| au ight| } [/latex]

Accepted Answer

From Eqs. 5.65

[latex]G_{0}=\int_{-\infty }^{\infty }{\left|G_{FF}(r) ight| dr}[/latex]

and 5.40

[latex]\sigma _{X}^{2}=G_{XX}(0)=G_{0}\int_{-\infty }^{\infty }{h_{X}^{2}(r)dr} \equiv G_{0}h_{x,var}(0)[/latex]

we see that choosing [latex] G_{0}=2A/c [/latex] will give the response variance of any linear oscillator with the delta-correlated excitation to be the same as the bound of Eq. 5.32

[latex]\sigma _{X}^{2} \leq h_{x,var}(0)\int_{-\infty }^{\infty }{\left|G_{FF}(r_{3}) ight| dr_{3} }[/latex]

for the response variance for [latex] \left\{F\left(t ight) ight\} [/latex] excitation. Of course [latex] \left\{F\left(t ight) ight\} [/latex] is nearly delta-correlated only if [latex] c [/latex] is large, and we can also note that [latex] G_{FF}( au ) [/latex] can be considered to tend to [latex] (2A/c)\delta ( au ) [/latex] in the limit as [latex] c ightarrow \infty [/latex]

Question 5.9: Choose Go for a delta-correlated approximation of a stochast...

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