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Jointly WSS

  1. Jan 13, 2013 #1
    1. The problem statement, all variables and given/known data

    ϕ(t) = R(t) = autocorrelation
    ϕ(f) = S(f) = power spectral density
    Inputs v[t] and w[t] are zero mean, jointly WSS processes with auto-correlation se-
    quences ϕvv(τ ) and ϕww(τ ), cross-correlation sequence ϕvw(τ ), power spectrums ϕvv(ω)
    and ϕww(ω), and cross power spectrum ϕvw(ω). Let x(t) be the output of a LTI system
    h1(t) with input v(t), i.e. x(t) = h1(t)v(t), and let y(t) be the output of a LTI system
    h2(t) with input w(t), i.e. y(t) = h2(t)  w(t). Assume all quantities to be complex.

    (a) Show that x(t) and y(t) are jointly WSS. Determine an expression for ϕxy(τ ) and
    ϕxy(ω) in terms of the input statistics and the system parameters.

    (b) Is the cross power spectrum always non-negative, i.e. is ϕxy(ω) ≥ 0, for all ω? Justify
    your answer.

    (c) Specialize this formula to obtain an expression for ϕxx(ω).


    2. Relevant equations
    ϕxy(τ) = E[x(t)y(t+τ)]
    ϕx(f) = ϕv(f)|H|²
    ϕx(f) = Fourier Transform( ϕx(t) )

    3. The attempt at a solution
    I'm at a complete loss. For jointly WSS, I need to prove x(t) and y(t) are WSS which they are because of v(t) and w(t) going through an LTI system. But I also need to show ϕxy(τ) only depends on τ.
    I feel that I'm missing a crucial point and don't know how to start.
     
  2. jcsd
  3. Jan 19, 2013 #2
    I suppose that it is trivial to show X(t) and Y(t) are WSS :wink: So the problem is only to prove X(t) and Y(t) are jointly WSS.
    To do this, expand [itex]R_{XY}(t,t+\tau)[/itex]:

    [itex]X(t) = \int_{-\infty}^{+\infty}V(t-\alpha)h_1(\alpha)d\alpha[/itex]

    [itex]Y(t+\tau) = \int_{-\infty}^{+\infty}W(t+\tau-\beta)h_2(\beta)d\beta[/itex]

    Hence:

    [itex]R_{XY}(t,t+\tau) = \int_{\alpha=-\infty}^{+\infty} \int_{\beta=-\infty}^{+\infty} E[V(t-\alpha)W(t+\tau-\beta)]h_1(\alpha)h_2(\beta)d\beta d\alpha[/itex]

    It is easy now :)
     
    Last edited: Jan 19, 2013
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