[Numpy-discussion] def of var of complex

Robert Kern robert.kern@gmail....
Tue Jan 8 20:56:10 CST 2008

Neal Becker wrote:

> 2 is what I expected.  Suppose I have a complex signal x, with additive
> Gaussian noise (i.i.d, real and imag are independent). 
> y = x + n

Not only do the real and imag marginal distributions have to be independent, but 
also of the same scale, i.e. Re(n) ~ Gaussian(0, sigma) and Im(n) ~ Gaussian(0, 
sigma) for the same sigma.

> Consider an estimate \hat{x} = y.
> What is the mean-squared-error E[(y - x)^2] ?
> Definition 2 is consistent with that, and gets my vote.

Ah, you have to be careful. What you wrote is what is implemented. Definition 2 
is consistent with this, instead:

   E[|y - x|^2]

But like I said, I see no particular reason to favor circular Gaussians over the 
general form for the implementation of numpy.var().

Robert Kern

"I have come to believe that the whole world is an enigma, a harmless enigma
  that is made terrible by our own mad attempt to interpret it as though it had
  an underlying truth."
   -- Umberto Eco

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