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analytical problem to find auto-correlation for a state space system

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upinderjeet
upinderjeet 2014-10-7
This is the problem from the topic of mean and covariance propagation in a state space system with random noise inputs
Consider the (scalar) system
˙x(t) = αx(t) + w(t),
where E[(x(t1)− ¯x)(w(τ )− ¯ w)] = 0 for all τ ≥ t1,
E[w(t)w(t + τ)] = qδ(τ),
and E[w(t)] = ¯w.
a) Find α such that the mean of the resulting stationary process is ¯x = c. (Hint: This is the steady state of the differential equation of the mean.)
b) Using a time domain approach, find the autocorrelation RX (t1, t2) = E[x(t1)x(t2)] of the resulting stationary (steady-state) process.
I have found the answer for part (a) and that is alpha=0. But I need help for part (b).

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