Suppose {Nt:tR} is a zero-mean white Gaussian process which is the input to a linear time-varying system with impulse response

h(t,)={n=1Nfn(t)fn()for0t1,010otherwise

where f1,f2,...,fN are functions satisfying 01fn(t)fm(t)dt=nm . Here nm is zero when n=m, it is 1 when n = m.

Let {Xt;tR} denote the resulting output process (i.e., the output process when the input N_t, goes through the linear system h(t,) )

(a) Find the Karhunen-Loeve expansion of {X_t;t [0, 1]}. That is, fifind an expansion that would

write the process as a sum product of sequence of functions of time and random variables.

Xt=nn(t)Zn

(b) What is the joint distribution of any two of the coefficients (i.e., random variables) in the expansion? That is, f_ZnZm(u, v)

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