[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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