Expectation value of infinite sum
The question:
Consider a stochastic process in which N(0,1) Gaussian white noise Zn is filtered to obtain Xn by an IIR filter : xn - ϕxn-1 = σzn
a) Expand this recurrence relation into an infinite sum xn = Σ∞k=0 akzn-k
Use this infinite sum to compute E[Xn]. Also compute the variance of Xn.
Where I'm stuck:
The requested infinite sum is xn = σ[Σ∞k=0 ϕkzn-k] but I'm having trouble grasping how to determine the expected value E[Xn]. I understand that the expected value is the average value of the series, but E[Xn] = Xn / n seems too simplistic. What I've found on the internet doesn't really clarify things for me, and the notes and lectures for the class leave a lot to be desired.
Can anyone explain how to get E[Xn]?
Thanks.
The question:
Consider a stochastic process in which N(0,1) Gaussian white noise Zn is filtered to obtain Xn by an IIR filter : xn - ϕxn-1 = σzn
a) Expand this recurrence relation into an infinite sum xn = Σ∞k=0 akzn-k
Use this infinite sum to compute E[Xn]. Also compute the variance of Xn.
Where I'm stuck:
The requested infinite sum is xn = σ[Σ∞k=0 ϕkzn-k] but I'm having trouble grasping how to determine the expected value E[Xn]. I understand that the expected value is the average value of the series, but E[Xn] = Xn / n seems too simplistic. What I've found on the internet doesn't really clarify things for me, and the notes and lectures for the class leave a lot to be desired.
Can anyone explain how to get E[Xn]?
Thanks.
Last edited: