Manuscript Psy02-B@2 · Psy02-B-v2.tex · section · cfd06e2e

Bath covariances and filters

Track A3: concentration and temporal fluctuations › Bath covariances and filters · lines 710–728

This convention is consistent with standard noise analysis Clerk. Equal-time variance alone does not determine the frequency placement or filtered variance.

For the memoryless carrier, put $r = κ + γ + R$ . In its specified Markov dilation, the phase-compensated $X_{t}$ has

E X_{t} = e^{- r t}, \frac{1}{2} E {X_{t}, X_{s}} = e^{- κ m i n (t, s)} e^{- r \abs t - s} .

Consequently

C_{X} (t, s) = e^{- κ m i n (t, s)} e^{- r \abs t - s} - e^{- r t} e^{- r s} .

Displayed formulas in this passage

unlabeled · line 718
$E X_{t} = e^{- r t}, \frac{1}{2} E {X_{t}, X_{s}} = e^{- κ m i n (t, s)} e^{- r \abs t - s} .$
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$C_{X} (t, s) = e^{- κ m i n (t, s)} e^{- r \abs t - s} - e^{- r t} e^{- r s} .$

Nearby in this section

section: Bath covariances and filters
theorem: Theorem: Exact filtered variance
proposition: Proposition: Windowed carrier variance
section: Bath covariances and filters