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Correlation of Complex Random Variables

  1. Jul 5, 2011 #1
    Hi,

    Why there is a half factor in the definition of the correlation of complex random variables, like:

    [tex]\phi_{zz}(\tau)=\frac{1}{2}\mathbf{E}\left[z^*(t+\tau)z(t)\right][/tex]?

    Thanks in advance
     
  2. jcsd
  3. Jul 6, 2011 #2
    I don't think that's true as a general rule. For the example you give, an autocorrelation, the general formula would be

    [tex]\rho_{zz}(\tau)=\frac{\mathbf{E}\left[z^*(t+\tau)z(t)\right]}{\mathbf{E}\left[z^*(t)z(t)\right]}[/tex]

    I'm guessing that in your case, 1/2 is just the normalization factor 1/E[z*z], perhaps because the real and imaginary parts of z are independent with mean square 1.
     
  4. Jul 6, 2011 #3
    does this general formula apply to the real-valued case, too?
     
  5. Jul 6, 2011 #4
    Yes.

    The general formula for a correlation is [tex]\frac{Cov(x,y)}{\sqrt{Var(x)Var(y)}}[/tex]. In the case of an autocorrelation, x, and y are the same (except displaced in time, which doesn't affect the variance), so the denominator reduces to Var(x) = E[x^2].
     
  6. Jul 6, 2011 #5
    So, 0.5 is just a normalization factor. Ok thanks a lot.

    Regards
     
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